I get the sense 200 years is how long it took to recover to a point where nation-states are viable again. Based on in-universe text, the majority of the population died in the opening days of the nuclear exchange and much of the continent was left uninhabitable or ravaged by freak mutant animals that survivors or their aimless descendants couldn't deal with. Not to mention the generational ennui of being the descendants of a civilization that would never be equalled in their lifetimes no matter how hard they tried.

There's thriving settlements again by the time of FO1 which means at least a feudal stage, and the NCR reportedly has industrial manufacturing in some capacity by NV. The Legion only exists because Eddy Sallow rediscovered Rome and implemented a version of it himself.