Is this a fundamentally relativistic phenomenon? 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? 

 A: There is a clear conceptual explanation that happens all the the frame of one small train car.
The idea is that the operator of each car is handed instructions telling them when according to their clock to fire rockets on which parts of their car.
And they follow the instructions. And the instructions are handed to every single car. Clocks are synchronized and then the instructions get followed. When the instructions are labelled "Einstein simultaneous acceleration profile a(t)=blah, car #508" the person in the car is astounded to find out the rockets first fire at the same time, but their instructions say they are instructed to fire rockets harder on the front of the car before they fire the rockets harder on the back end of the car. These instructions, when followed, stretch out the car as well as generating an acceleration. And the mismatched timing of the increased thrusts rips the car apart.
They will think it was strange the instructions were labelled with the name "simultaneous acceleration" when they had to accelerate the front part before the back part. But the labels your boss sticks on your instructions isn't a physical cause. The physical cause is the rockets ripped the car apart.
The only place relativity came up is when you decided you wanted to have each car fire thrusters so that the whole thing accelerated in a way that was simultaneous to the inertial observer. But without relativity, no one would hand you those instructions to follow. So you wouldn't do the experiment, so you wouldn't observe the phenomena.
And if it's so vague to just say that there is some velocity at which cars break and it doesn't predict the velocity then it's not falsifiable.
A: As Timaeus says this is another version of the Bell's spaceship paradox and as such has been discussed many times over the years. Let me suggest a way that seems to me to clarify what is going on.
Consider two observers on the train, $A$ who is at the origin at time zero and $B$ who is some distance $d$ along the train at time zero. If the train is accelerating with constant proper acceleration $a$ then the positions of the observers in the track frame as a function of track frame time are given by:
$$\begin{align}
 x_A(t) &= \frac{c^2}{a}\left(\sqrt{1 + \left(\frac{at}{c}\right)^2} -1 \right) \\
 x_B(t) &= x_A(t) + d
\end{align}$$
This is a standard result that you'll find in e.g. chapter 6 of Gravitation. As you say in the question the spacing between the observers is constant in the track frame.
Now let's switch to the rest frame of observer $A$. The key thing you need to know is that for an observer with constant proper acceleration $a$ their spacetime geometry is described by the Rindler metric:
$$ ds^2 = -\left(1 + \frac{ax}{c^2}\right)^2c^2dt^2 + dx^2 \tag{1} $$
Proving this is straightforward but tedious so rather than do it here I'll just refer you to the first hit that came up when I Googled it.
For our purposes the key feature of this metric is that it predicts there is time dilation comparable to that you'd find in a gravitational field. If we take $dx=0$ and use the fact that $ds^2 = -c^2d\tau^2$ equation (1) becomes:
$$ \frac{d\tau}{dt} = 1 + \frac{ax}{c^2} $$
where $t$ is the time measured by observer $A$ and $\tau$ is the time measured by an observer at the position $x$. So in our scenario $A$ observes $B$'s time to be dilated by a factor of:
$$ \frac{dt_B}{dt_A} = 1 + \frac{ad}{c^2} $$
I use the conventional term dilated, but actually $B$'s time runs faster than $A$'s. This matters because that means the acceleration of $B$ measured in $A$'s frame, call this $a_B$, is greater than the proper acceleration $a$ by a factor of $(dt_B/dt_A)^2$:
$$ a_B = a\left(1 + \frac{ad}{c^2}\right)^2 $$
So even though $A$ and $B$ have the same proper acceleration, $A$ observes $B$ to be accelerating away at $a^2d/c^2$.
And that's why the train stretches.
A: 
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.)

If I understand the above correctly, you're specifying that each point of the train car has the same coordinate acceleration as observed from the inertial reference frame (IRF) of the track and so, the world-lines of the points of the train car are congruent.
Now, in SR, an object cannot have uniform coordinate acceleration since, in that case, the speed of the object would eventually reach and then exceed $c$.  So, let's further stipulate that the coordinate acceleration of the points of the train is of the form
$$a(t) = \alpha \left(1 - \frac{v^2(t)}{c^2} \right)^{3/2},\qquad t \ge 0$$
where $v(0) = 0$.  That is to say, the acceleration begins at $t = 0$ and each point of the train car has constant proper acceleration $\alpha$ in the rest frame of the track.
It's easy to see with a spacetime diagram that, for $t \ge 0$, and in a momentarily comoving reference frame (MCRF) of any point of the train car, other points along the length of the train car have different velocity and proper acceleration - the points more forward are moving faster and their accelerometers read larger numbers while the points more rearward are moving slower and their accelerometers read smaller numbers. 
In other words, there is no MCRF for the train car as a whole; from a MCRF of a point, the train car is expanding along the direction of the acceleration.
Note that the situation is vastly different in Galilean relativity where there is a MCRF for the train car as a whole.

But there is a limit to how much you can stretch a train car, so
  beyond some velocity the train must snap.

Not according to your setup.  You've stipulated that "each point on the train car experiences the same positive acceleration a(t) at each time t".  
Since this is the case, the train car does not snap by stipulation.  Yet we know that there are MCRFs in which the ends of the train are moving with vastly different velocities and have vastly different accelerations which is clearly not physically reasonable.
So, it is this stipulation that you must closely examine.  That is to say, one cannot observe the points of the train to have the same constant proper acceleration.
As another answer has pointed out, for the points of the train to maintain constant local distance in a co-moving frame, the points in back must have greater proper acceleration than the points in front.
A: Here is my answer in words attempting to translate the math in the other answers. 
First, 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.

should be more accurately stated as
The Curious (But Not So Much Because Difficult To Set Up) Phenomenon:
If one manages to accelerate each car (using a rocket probably) in a train car such that at very high velocity those cars are accelerated uniformly as seen from the train station, then the train car must snap. 
It will snap because the rocket in a given car has to be more powerful than the one behind, else we cannot observe the uniform acceleration of cars from the train station.
To say it otherwise: if the train is simply accelerating like a proper train, all cars following the lead engine, then when relativistic velocities are reached each car will not have the same acceleration from the point of view of the train station.

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?

It is relativistic because it is only at relativistic velocities that one cannot accelarate a train car and get uniform acceleration along the cars (as seen from the train station) from the train engine alone.

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?

We would probably never observe something similar in nature: how and why would a extended compound natural system arrange the behaviors of its longitudinal components in such a way that those components mutual spacings appear longitudinally stable in a referential relatively to which they are permanently accelerating?
