In defense of 200 years "exactly": Word of Todd Howard, Pete Hines, and Codsworth, and robots are apparently "comically precise"; this paragraph is smaller, but these details are still heavy enough to throw it into question.
In defense of probably being more than 200 years: Bethesda's MO has always been to move the timeline forward, even to the point that they specifically told Obsidian that New Vegas had to follow suit; as far as we now, they haven't changed that philosophy to put it before New Vegas. Also, the advanced state of the Brotherhood implies that they've had more than just a few months to assemble vertibirds, an airship, and head to the Commonwealth, not to mention the safe guess that the Prydwen (named after King Arthur's ship) is led by Arthur Maxson, now an advlt. And as for Codsworth's 200 years line, there's a "200 years?" follow-up dialog choice to that, where Codsworth would presumably be more precise. If he was being comically precise, he would have mentioned the months, days, and minutes in addition to years, wouldn't he?
But honestly, I don't care any more. We've all derailed enough threads, haven't we?