This question was inspired by some silliness in other threads but is independent of that silliness.
Say that a train car sitting on a track is accelerated uniformly along its length if each point on the train car experiences the same positive acceleration $a(t)$ at each time $t$ as measured from the track's frame. (The track is not accelerated --- it remains in the same inertial frame.)
Clearly such an acceleration cannot change the length of the car in the track's frame, so its proper length (which has to be greater than its length in any other frame) must increase. That is, an observer on the moving car must say that the car has stretched. But there is a limit to how much you can stretch a train car, so beyond some velocity the train must snap. The snap should be observable to anyone, including an observer stationary with respect to the track.
Therefore we have what I will call the Curious Phenomenon:
If a train car reaches a sufficiently high velocity as a result of being accelerated uniformly along its length, then the train car must snap.
Note that the statement of the Curious Phenomenon (as opposed to the derivation of that phenomenon) has nothing to do with relativity. Note too that the phenomenon is in principle (though perhaps not in practice?) directly observable.
This leads me to two questions, which might or might not be the same question in disguise:
Question 1: Is there a clear conceptual explanation of the Curious Phenomenon based on classical mechanics without invoking relativity? Or does one really need relativity to explain this?
Question 2: Suppose we knew nothing about relativity, but had observed the Curious Phenomenon. Would the search for an explanation naturally lead to relativity in the same sense that say, a search for an explanation of the Michelson-Morley phenomenon could naturally lead to relativity?