If the signal is severely clipped, then yes; the extra power induced can lead to thermal failure of the voice coil. But one can't just automatically assume that he blew his sub because he was running 250W to it instead of 400W. Sure, it's possible if he clipped the piss out of his amp... but there are other reasons as to why the driver would fail.
It could be that his box is way too big, or it's a ported enclosure and he played something way below tuning -- and seized it up from over-excursion. He hasn't really elaborated.
OP -- If you really want to know how this happened, you need to give us more info. Be thorough, specific, and honest -- we aren't going to buy any BS lines about how you were just playing some Barry Manilow at low volume... what did you have cranked right before you locked it up tight?