The Hume-Edwards-Campbell (HEC) principle says that if you have a bunch of items, and each one is explained, then the whole bunch might be explained. In particular, any infinite regress might be a complete explanation. The Hume-Edwards principle replaces the "might" with "is". I've published counterexamples to the HEC before, but here is a cool recipe for generating counterexamples.
Let p1,p2,... be an explanatory regress of propositions, so p2 explains p1, p3 explains p2, and so on. Suppose (as might easily be the case) that there is some proposition q such that (a) q couldn't be self-explanatory, and (b) the pi are all clearly completely explanatorily irrelevant to q. Now, let qi=pi&q. Then q1,q2,... are an infinite explanatory regress. But if q couldn't be self-explanatory, this regress can't be completely explanatory as it does nothing to advance the explanation of q.
I might have got the basic idea here from Dan Johnson. I can't remember. The counterexamples depend on the idea that if A explains B and Q is irrelevant, then A&Q explains B&Q. I am a bit less sure of that than when I started writing this post (which was quite a while ago).