Given the high infrastructure cost for a railroad system, I don't think that an airline can afford to run one on its own. Sure, there might be Virgin Trains in the UK, but they use the existing infrastructure and only pay a "utilization fee" to the railtrack company. Since there are no real highspeed tracks in the USA (with "real" I mean for speeds of 250-300km per hour), the whole infrastructure would need to be planned, built and paid in the first place.
However, what I could very well imagine, is that airlines do codeshares with train systems, e.g. on the New York-Washington route. If I recall right, the trains even stop at BWI airport?
In Europe, for example Lufthansa and KLM have done this. LH operates a dedicated waggon in the normal German rail Intercity Express train from Stuttgart to Frankfurt. Passengers check in at the railway station, get their boarding pass for the onward connection ex FRA and the bags are also already checked through. The traveling time by train is about 65 minutes. A flight takes maybe 25 minutes, plux taxiing, waiting before boarding, etc, so would not be faster.
So who knows, maybe we will see an Amtrak train with a Delta codeshare soon in New York Penn station??