The following I sentence should be odd I think on most theories of how epistemic modals work.

(5) ?I might know that Clinton was president.

And it does sound pretty bad to me. The problem is that it’s very hard to really know something if you’re in a position to represent yourself as being uncertain about whether you know it, and in saying "It might be that p", you represent yourself as having some uncertainty about whether p. The representation here is an implicature, I think, not a saying. If I say I might be in Providence, that’s true, even though there’s no doubting the matter. But it’s an odd thing to say for just that reason.

OK, no problems so far. But the theoretical argument here shouldn’t be affected by whether we use a that clause or an embedded question. But it is! (6) is not problematic, in the right contexts.

(6) I might know where we can get some beer.

(Imagine saying this in a conversation about where to get late-night beer in a strange city without obvious beer supplies.)

Google reveals that this kind of locution is in fact quite common. What’s even more bizarre is that it’s common to qualify even more, as in (7).

(7) I think I might know where we can get some beer.

No theory here, just an observation.