Two years ago I wrote a post geared towards a refutation of the existence of brute facts and what this entailed for naturalism. Since this time I have engaged in many discussions with naturalists regarding this very topic, and as a result of those discussions I have (slightly) altered and polished my argument. And because of this I have, for a while now, wanted to write up another, more systematic, post which attempted to demonstrate the impossibility of brute facts. So, here goes.
The Stanford Encyclopedia of Philosophy defines a brute fact as synonymous with an unexplainable fact.  In the same vein, Wikipedia states, "a brute fact is a fact that has no explanation. More narrowly, brute facts may instead be defined as those facts which cannot be explained (as opposed to simply having no explanation). " Right away we see that we can distinguish between two types of brute facts--those in practice (extrinsic), and those in principle (intrinsic). An extrinsic brute fact is a fact that we currently do not have an explanation for. An example of this would be the origin of life. We believe that the fact of life's origination has an explanation, we simply do not know currently what that explanation is. On the other hand, an intrinsic brute fact is a fact that has no explanation, at all, in principle. Considering an intrinsic brute fact, it's not that we don't know the explanation for X, but that X doesn't have an explanation to be known in the first place.
Now, nobody disputes the existence of extrinsic brute facts; we all agree that there are things we don't know. The point of contention, and our domain of focus in our discussion of brute facts, is whether or not intrinsic brute facts actually exist or whether they are even possible. This is what is to be considered presently.
What should be noteworthy at this point is the notion of a brute fact is defined in terms of explanation. And, as we'll see below, the concept of explanation, and what it entails, is crucial in determining whether or not brute facts are possible. So, what constitutes an explanation? Well, the problem with defining this term is that an explanation can be given in many different domains and contexts. Take these different cases of explanations: the teacher explains the lesson; Susan explains what she meant; the nail explains why the tire is flat; the nonzero net force explains the change in velocity; the premises explain the conclusion etc. Notice that in each of these cases the presence of something--e.g. someone doing the explaining, an object, a force, an abstract concept or proposition--is the reason for, and clarifies, that which, in the absence of that something, would be unclear--the lesson, the meaning, the flat tire, the change in velocity, the conclusion.
The important concepts just utilized are reasons and intelligibility. When some fact is explained, there is a reason, account, or justification--which can be grounded in many things, from an object to an abstract proposition, as we saw above--which imparts some form of clarity and intelligibility to the thing that was heretofore unclear. Therefore, I maintain that the proper definition of "explanation" that should be utilized is "a reason whereby something is rendered intelligible." 
Now, from this definition is entailed a crucial inference: something being explained is not the same thing as something actually having an explanation. The former is an action, dependent on minds doing the explaining, while the latter is not. Return to a couple of my examples above--namely, the teacher explaining the lesson, and the premises explaining the conclusion. The teacher explaining the lesson is an example of something actually being explained, and we could call this a case of explanation in practice (extrinsic). The premises explaining the conclusion is an example of something, the conclusion, having an explanation while not necessarily being explained to anyone currently, and we could call this a case of explanation in principle (intrinsic). (Note at this point that there is an identical differentiation here between the different types of explanation and brute facts.)
Not only are intrinsic explanations not equivalent to extrinsic explanations, but the latter actually presuppose the former. Intrinsic explanations are a necessary condition for extrinsic explanations. That is to say, in order for one to be able to explain something, there has to be an objective explanation available in the first place. It is incoherent to claim that something was explained that had no explanation. The teacher cannot give reasons that render a lesson intelligible if the lesson doesn't have a reason for its intelligibility. This is important because many of my interlocutors have claimed that explanation should only be focused on explanation in practice, yet this is nonsense, since, to reiterate my point, explanation in practice presupposes harboring an explanation in principle.
Another concept that is embedded in the notion of explanation is that of intelligibility, and just like explanation and brute facts, it comes in practice and in principle. For something to actually be made intelligible in practice means that someone has actually comprehended it. For something to be intelligible in principle means that it's possible that it could be comprehended, even if it never actually is. For example, the mechanism of gravity was unintelligible in practice for many decades, even though it was always intelligible in principle--that is, there always was an account for the mechanism of gravity. And once again, just like explanation, intelligibility in principle is a necessary condition for intelligibility in practice. For in order for someone to comprehend X, it is a necessity that its possible that X be comprehended in the first place.
Now, since we're currently interested in intrinsic brute facts, and not extrinsic brute facts, this entails that we are likewise interested in the nature of intrinsic explanation and intrinsic intelligibility, and not extrinsic explanation and extrinsic intelligibility. For since we're attempting to determine whether facts can be unexplainable in principle, we need to examine the nature of explanation in principle and see if this is something that can be done away with whilst a fact remains intelligible in itself. We will not go the way of the PSR and merely dogmatically assert that all things must have an explanation, rather we need only delve deeper into the nature of explanation and infer its entailments.
To begin, let us consider an example, similar to the example I utilized in my original post, of explanation in principle. In physics acceleration represents the rate of change of velocity with respect to time. Now, if we are curious as to the explanation of acceleration in principle, we can find this in the concept of velocity, since acceleration is literally defined in terms of velocity. And if we go further and ask for the explanation of velocity, we find this in the concept of the rate of change of position. Therefore, acceleration is explained by velocity which is in turn explained by position. What this means is that acceleration is granted intelligibility by the concept of velocity, which is granted intelligibility by position, so on and so forth.
This entails something very important: acceleration is ultimately explained by position. In the specific explanatory chain that we are considering, all the concepts that lead up to position are only ultimately imparted their intelligibility by position itself. If position is dropped, then so are all the subsequent concepts--if position is rendered unintelligible, then so is acceleration.
In any event, the notion that should be highlighted here when speaking of essentially ordered series is that of derivation or of "being imparted". That is, when A explains B, this means, per our definitions, that B is rendered or imparted intelligibility by A, that B would be unintelligible were it not for A. To return to our example, acceleration would be unintelligible were it not for velocity and thus acceleration derives its intelligibility from velocity. But, again, this is the case for any subsequent members of an explanatory chain with regards to a specific member. That is to say, if A explains B which explains C which explains D, then D ultimately derives its intelligibility from A, and only proximately derives its intelligibility from B and C.
What this likewise entails is that if A does not impart intelligibility to B, then B does not impart any to C, and likewise for D. And this is where consideration of brute facts come in. For where would a brute fact fit in such a chain of explanation? Surely it cannot be the highest member of an explanatory chain, because since a brute fact has no explanation then, by definition, it has no reason whereby it is rendered intelligible, and thus it would be unintelligible. But a reason needs to be intelligible itself if it is to render something else intelligible. (Again, consider acceleration and velocity, the latter has to be intelligible if it is to impart any intelligibility to acceleration.) Therefore, A being intelligible is a necessary condition for A to be an explanation for any B. Logically, this means that if A is unintelligible then it cannot be an explanation for any subsequent fact B. Thus, a brute fact cannot be the first member in an explanatory chain, for it would not explain anything while being itself unexplained. 
However, while a brute fact might not be the highest member in an explanatory chain, is it possible that it be a member somewhere in the middle? That is, can we have a chain A, B, C, D...wherein C is a brute fact? I don't see how, since, to reiterate the above point, C would not be able to explain D, and C could not be subsequent to B since B would then have to explain C--otherwise C wouldn't be in the chain to begin with--which would contradict our original premise for the nature of C--namely, its having no explanation. What we see, then, is that there is no place in an explanatory chain for a brute fact, and if there is no place in an explanatory chain for a brute fact then it would seem that brute facts are impossible.
In addition to dispelling the notion of brute facts participating in the nature of explanation, we can add another argument--call it the argument from intelligibility. The argument can be formulated as follows: If a member F of an explanatory chain M is intelligible, then no members antecedent to F in M can be brute facts. Why can we deduce this? Well, if F is intelligible then it has a reason whereby it is rendered intelligible--otherwise it wouldn't be part of an explanatory chain, since it wouldn't have an explanation. And since intelligibility is imparted down through the members of explanatory chains, we can say that for F to be intelligible, every member of M must also be intelligible--otherwise intelligibility is not imparted at some point in the chain. But if every member is intelligible, then every member must have a reason whereby it is rendered intelligible--again, otherwise it wouldn't have an explanation. And this means that every member does in fact have an explanation and cannot be a brute fact. Based on this line of argumentation, we can reach an interesting conclusion: if a fact is intelligible, then it cannot in any sense be linked to a brute fact, and thus we cannot posit a brute fact from anything that exists.
Now, attentive readers may have noticed something that it seems I have forgotten: namely, that there can be multiple explanations for something, and that these multiple explanations can form multiple explanatory chains that are interconnected, and thus, it would seem, my account of explanation and explanatory chains is too simplistic. I do agree that something can have multiple explanations and that there can be multiple interconnected explanatory chains that stretch across different domains. However, I maintain there is nothing about multiple explanatory chains that changes the nature of an explanatory chain in itself--and thus there is nothing that calls my conclusions into question.
To illustrate this, consider a case of combustion, perhaps a candle that is lit. What is the explanation for this lit candle? Well, we can think of a couple. One explanation is the fact that somebody actually lit the candle, from. Another explanation is an oxidizing agent and a chemical reaction. These are both genuine explanations since they are both reasons wherein the lit candle is rendered intelligible. And here we also have two explanatory chains that converge on a single state of affairs. So, we have a situation where one explanatory chain (...A, B, C) converges with another chain (...X, Y, C) at the fact of C--the lit candle.
The question that should be considered presently is whether or not the existence of C changes the nature of explanation considered above. It's difficult to see how it would change everything we've considered, since we still have something (C) which derives its intelligibility from antecedent members, such that, in the absence of such members, it would be rendered unintelligible. The only "new" notions that need be introduced are partial intelligibility and partial explanations. For if C is missing one explanation, out of two, then it is only partially intelligible, and thus the explanation it has is only a partial explanation. But notice that none of this changes the nature of explanation itself. It is still the case that in order for C to be intelligible in any sense, it needs to have at least one reason whereby it derives its intelligibility--even if this is only a partial intelligibility. And, more importantly, the fact that C derives its intelligibility from something else means it has to, at least, be a member of an essentially ordered explanatory series, which means that all our conclusions from above still hold true.
To substantiate this even further, consider the question at the forefront of this post: can a brute fact be part of a convergence of multiple explanatory chains? This doesn't seem possible since, again, a brute fact cannot have an explanation, by definition, and therefore it cannot have antecedent explanatory members. That is to say, a brute fact cannot be reliant upon another fact for its intelligibility, much less multiple facts. Furthermore, based on our intelligibility argument above, what we can also say is that if C is intelligible, then, even if it is part of multiple interwoven explanatory chains, we can safely say that none of the antecedent members in these chains contains a brute fact.
In summarization we've concluded many things. First, because of the nature of explanation itself, as well as essentially ordered explanatory series, brute facts are simply impossible. There is, logically, nowhere they can fit in chains of explanation, and a chain of explanation is the only place they would go if they were possible, since brute facts are defined in terms of explanation. Second, because of the nature of explanatory chains, if a fact or state of affairs is intelligible, then it follows that it cannot be associated with a brute fact. Third, no matter how interconnected and interwoven multiple explanatory chains are, they still retain their nature as essentially ordered series and our notion of explanation--with which the whole post is founded on--and our conclusions regarding the impossibility of brute facts remain intact.
 See the article on the Principle of Sufficient Reason
 Let it be noted that this is one of the most crucial points in my argument. For everything from here on out follows from, and is entailed by, this definition.
 Note that neither can a brute fact be the lowest member in an explanatory chain, since in order to be the lowest member, some member would come before it and this member would have to explain it, which would contradict the definition of a brute fact. Yet I don't think anybody would entertain this idea since usually a brute fact is seen to occur at the beginning of a long line of explanations, and not at the end.