Well, I just googled my own claim, and discovered that Feynmann says that the story that scientists were worried about coming to the end of physics is apocryphal. Apparently they were well aware that there were serious problems with classical mechanics from 1850 onwards. Feynmann wasn't often wrong about anything in science, so I'll await your answer with interest.
I thought it was someone more famous, but I'll have to do some of my own research.