I think that one of the problems on these types of discussions is that all the focus seems to aimed at the voice coil (overexcursion, heat dissipation); another important factor contributing to over excursion is cone suspension fatigue, after a while due to either improper drivers or use; cone edge fatigue takes place, the surround becomes soft allowing too much movement resulting in overexcursion.
Absolutely possible and probable over the lifetime of a driver. As the suspension wears, of course its compliance increases. The parameters of the driver change over time. This will also cause the driver to interact differently with its enclosure.
My second failure mode made no specific reference to the voice coil. Overexcursion, as you and Bill have said, can have a variety of effects on a driver.
Crockettnj, Bill already covered your comment, but I'll state that I strongly agree with him. The shape of a waveform will not damage a driver. If a driver cared about the acceleration of its cone, would it not be damaged simply by supplying it with a higher frequency signal? A woofer goes into cone breakup modes to reproduce higher frequencies, though they're very directional. Above that, it simply doesn't reproduce significant signal.
Feed a square wave (or something close to that - any signal with a lot of high frequency content from a distortion box, or from excessive clipping over a power amp) to a multi-driver cabinet (ie a 2-way bass cabinet). The woofers will reproduce what they're capable of (for the sake of argument 2khz) and won't have any appreciable output beyond that in any direction. The harmonic content above that will be handled by the high frequency driver(s).
Your drivers probably blew because of overpowering or overexcursion. If the enclosure wasn't very well suited to the replacement drivers, they could very well have exceeded Xmax or Xmech as you played. Did you ever take the driver apart to examine what caused it to fail? It's pretty easy to identify overpowering or mechanical failure by seeing the damage to the coil/former.