(This question might be too vague, feel free to edit or vote for closing.)

   In math there are usually lots of non-existence theorems. When someone presents such a theorem, one natural response is "why shall I even care", or "why should such a thing be impressive". 

   The problem is, in the case of a non-existence theorem, usually all examples are trivial. If you tell some undergrad non-constant bounded entire functions don't exist, he/she will probably reply with a shrug. Similar thing happen to me and my friends when we talk about some fancier theorems (or when I see a paper stating such a non-existence theorem). I feel like it's really hard to convince people (or convince myself) "a priori this thing could exist, however by this awesome theorem it doesn't."

   I think people would be impressed if all hypothesis look innocent, like the one in Liouville's theorem on entire functions, or maybe the fact that people have seen the existence of differentiable bounded non-constant functions helps. In general it is not necessary that all hypothesis look friendly. How would one figure out whether a non-existence theorem is a good one or is true just because one of the hypothesis is insanely strong?