I live in Northern Virginia, but travel to NYC enough to where it makes me ask, which mode of travel - car, train, or plane will get me from my door to Times Square the fastest. It seems fairly even, but I wonder how far one needs to travel for the plane to beat the train
A lot depends on how far you live from the airport or train station - I have a much larger "train is faster" radius because the airport is on one side of "the big city" and I am on the other - and there are train stations on my side.
The big difference is in the time to clear security and wait at the gate. I have read suggestions you arrive at the airport 3 hours before your flight. For domestic flights I go with 2 hours, even less when I know I can fast-track the lines, but that's up to you. For the train, you just need to get there before the train. The second difference is that airports are usually 30 minutes or more from downtown, while train stations are downtown - depending on where you're headed, that can be a big deal. I count that as another hour's time in the train's favour. And finally, there is a fair amount of "ceremony" in the actual departure and arrival of a plane - safety speeches and checks before departure, taxi-ing, waiting for takeoff, possible circling of the airport on arrival, more taxi-ing, waiting for the ground crew - that can add up to another hour.
For me, the train has a five hour headstart on the plane, because on top of all that I can get to the train station quicker than the plane, and it's already partway to my destination. That means I can go about 400 miles before the plane would be faster. This covers a fair amount of where I need to go, so I take the train quite often.
I often heard that for any station-to-station travel time under 3 hours, the train beats the plane and can capture most of the business travelers. This is supposedly the rule that was used when planning the early stages of the TGV train network in France (sorry, no authoritative source to back that up). This does seem to explain many design choices (building costly new lines on only part of the way to reach a key destination in time but not quicker, having some direct trains that do not stop at all before the final destination, etc.)
In the US I am not sure how much can be covered in that time but in Europe, at current train speeds, this means between 750 and 1000 km. For a plane, this translates to 1 hour in the air give or take but if you factor in check-in, security, walking in the airport, taxiing, travel to the airport and unavoidable waiting time in-between, the train can still be very attractive time-wise.
Small trivia: If you study maps carefully you might notice that many European train stations are in fact located along the limit of the densest core of the city in the 19th century. They are now fully enclosed in the city and much closer to the center than most large airports but still not exactly in the historical core of the city.