This is true for people. But one might expect robots to be a little more literal and exact and in the demo video doesn't the Mr. Handy remark something to the effect that the protagonist was 200 years late?
Still, I'm imagining it around 2284 simply because it's been seven years since Fallout 3 so I'm placing Fallout 4 seven years later in game as well. No deep theories or anything, just expecting a sense of continuity that wouldn't be present if it took place at the same time as the previous games.