I find people will often say something is impossible, when really they just aren't s mart enough to figure it out.

Physicists (and the Discovery channel) love pointing this out about time-travel: No law of physics, except for perhaps the law about increasing entropy, actually says that traveling backwards in time is impossible.

My favorite example of this: what's the minimum length of wire needed to connect the 4 corners of a unit square (square with edge=1, figure A below) to each other?   (The same section of wire may be involved in multiple connections, as demonstrated in figure B).

You may say, "That's obvious, the answer is 'C'". But can you prove it? Afterall, if you can't prove it, how can you really be so sure it's true? When I ask people, their proofs usually involve something like "it's so obvious, it can't possibly be any shorter". (If you know the answer, ask a smart friend who doesn't and you can see what I'm talking about)
It is interesting to see people passionately prove a wrong answer.

The flip side of this is when a problem is indeed provably-impossible, yet somebody insists on trying to find a solution. There are lots of math examples here too. And lots of Dilbert examples too.

The middle ground: And then there's a whole middle range. Perhaps you can't figure out a solution, but you can show that whatever it is, it must have some certain properties.