I've been musing—nothing I care about rides on this—about this kind of case. Let's say that I have ten independent pieces of evidence that Socrates was killed by the Athenians. They give me knowledge, but it's more evidence than I need—any seven or eight of them would be sufficient. Unbeknownst to me, one of the ten pieces of evidence turns out not to work (maybe it's a complete fraud, maybe it's Gettiered, maybe it doesn't support the claim that Socrates was killed by the Athenians—there are all sorts of ways of filling this one, depending in part on what you think evidence is). The other nine pieces are fine.
Now, if you ask me how I know that Socrates was killed by the Athenians, I will give you all the ten pieces of evidence as my story. But it seems I'll be wrong: it is false that the ten pieces of evidence give me knowledge that Socrates was killed by the Athenians—at best, nine of them do that. Moreover, I am unable to give the correct story about how I know, because I don't know which of the ten is bogus. Do I know that Socrates was killed by the Athenians?
I think so. So it seems that it is not necessary for knowledge that I be able to give a story about how I know. In this case I know, but I don't know how I know. Nothing earthshaking here. But I thought it was an interesting case, and probably pretty common.