I understand Newtonian mechanics fairly well.
I took courses in modern physics, Quantum and Relativity, and I could even work the homework problems, but I don't claim to understand it. I know all the stuff they talk about on the Discover channel, but I can't really explain it to anyone else, not properly, with all the math you really need. So I have tried to convince myself I had better not talk too much about it.
But it does work. Its one of the things I used to know, like how to field a ground ball and turn a double play.
As for Carl Sagan: nature doesn't care whether its a waste of time or not. Wise up, Carl. Nature doesn't have to make sense. That's our job.