In my experience, that's not the case. People (or at least people not primed to take a tricky-logic-puzzle stance) think they understand the problem just fine,

In my experience, that's not the case. People (or at least people not primed to take a tricky-logic-puzzle stance) think they understand the problem just fine, and quickly give the wrong answer. Asking about it afterwards (once you get past the defensive denials et al), it's usually revealed that they did in fact understand the problem. They just failed to grasp what a proof/disproof actually requires, because they've never had to think too deeply about the issue before.