In physics there a few no-X theorems that seem rather suspicious in the sense that they point to cracks in our current understanding of nature. One of them is Penrose's cosmic censorship hypothesis that translates to a "no-naked-singularity" conjecture. The most famous examples are at the center of black holes where there are supposedly singularities. But we will never know about them because they are shielded by black hole horizons (unless you are willing to kill yourself and fly inside the black hole, however you will not be able to report back to us). That to me sounds a bit suspicious.
Here is another one. The no-communication theorem in quantum mechanics. When two particles are entangled they become correlated with each other over arbitrarily large distances. This implies that if Alice measures the spin of one of the particle here at earth, it collapses the joint wave function, say in a spin-up state, then *instantaneously* Bob's particle at the other end of the universe collapses into a spin-down state. Special relatively forbids anything moving faster then the speed of light, because if it does for some observers a signal can arrive before it was sent! Similarly here: for certain observers Bob first checks his spin before Alice has measured hers and therefore this observer concludes that Bob collapses the wave function and not Alice!
The only way out is that these two interpretations are actually coherent and this means that no information can be send between Alice and Bob. Alice can not force the particle in a up state (or a down state for that matter) so that's no use. But perhaps she can transmit information by simply collapsing a superposition into one of the two pure states (irrespective whether it's up or down)? Bob only has to determine if the wave function is in state A+B or in one of the pure states A or B. Alas, Bob can not do this with a single measurement because his measurement will collapse the wave function into either A or B. Now what if he could make N copies of his wave function? Then by measuring all N copies he finds either that all of them are in state A or B or he finds some of them in A and some in B indicating that the original wave function was still mixed. Unfortunately for Bob, the no-cloning theorem comes to quantum mechanic's rescue which says that you can not make copies of wave functions.
Feeling uneasy? To me this seems like a theory that is trying to rescue itself. Not really the most concise explanation. We need multiple no-X theorems to wiggle ourselves out of difficult questions. What this points to in my opinion is that the current theory (quantum mechanics) is an unnatural (but still accurate) theory of nature. We reach the right conclusion but through weird complicated reasonings. Very similar to Ptolemy's model of the universe that made the correct predictions but was complicated and difficult to interpret. The new theory replacing quantum mechanics will hopefully act as Ocam's razor and bring natural explanations for quantum weirdness.