"I know that p, but I just can't get myself to believe it." On standard accounts of knowledge, this is self-contradictory. But I think the sentence gets at a very interesting phenomenon.
Here are two potential real life cases of the phenomenon, with names changed (we can't really discuss genuine real-life cases, because we can't read people's minds). George for years has been convinced by the apologetic arguments for Christianity. But he couldn't get himself to believe it. We would be tempted to say that he knew Christian doctrine to be the truth, but didn't believe it.
A second case is this. When we look at Dr. Schmidt's experiments at Auschwitz and try to explain his behavior, we are tempted to say: "His actions show that he did not believe Jews to be human beings." But in a juridical context, we are tempted to say something like this: "Moreover, the fact that he had no qualms about the application of the results of the experiments on Jews to non-Jewish Germans shows that he knew full well that Jews were human beings."[note 1] Odd, isn't it?
There are a couple of ways of understanding the phenomenon. One way is to say that sometimes we say "x knows p" simply to mean "x is in full possession of conclusive evidence that would suffice for knowing p". But I think this misses something in the phenomenon.
A different way, suggested to me by Bob Roberts, is that there are different senses of "belief" in play: one kind of belief—some kind of full-blooded, felt belief—is missing, and another kind of belief—the kind that "knowledge" requires—is present. One can parallel the two kinds of "belief" with two kinds of "knowledge".
Moreover, this issue connects, Bob suggested, with Socrates' doctrine that you necessarily do what you "know", a doctrine that requires a stronger sense of "know" than that which is used when we accusingly say that Dr. Schmidt knew the Jews he was experimenting on were human beings.
Maybe the stronger sense of "knowledge" is something almost visual (compare how Socrates in the Protagoras supposes that there is no knowledge when one is in the grip of a quasi-perceptual illusion, as when a temporally far-off bad seems lesser).