Verified answer

Its unlikely that a test exactly like that was carried out. But I do know there was an experiment where two atomic clocks were set to exactly the same time. One was put in a fast (supersonic) plane and flown to New York at a very fast speed. The other clock was sent to New York by a (far slower) boat. When the clocks were later unloaded and analysed, it was discovered that the clock on the supersonic plane was very, very slightly behind the other clock. So it is true that the faster an object travels, the slower time is for it. A fast trian would go through time very slightly slower than, say, somebody standing still. However, the difference would be neligible, and definitely unmeasurable, so no, this experiment never took place.

Note - By the way, if an objet travelled at the speed of light, time would freeze for it. If it travelled faster, time would go backwards. Sadly, travelling faster than the speed of light is impossible.

I don't think they've done it on a train (have vague memories about something involving atomic clocks on planes though), but the effect certainly is real, and often experienced. The easiest example to cite, but not exactly the same thing, is in GPS. Time flows at a different rate for the satellites simply because they're so far from the Earth, and the equipment has to compensate for this.