I'm sure others can chime in and cover the ones i've missed, especially on the mechanical limits.
Thermal limits:
Too much power to the point where the coil cannot handle the heat. Clipping in the signal causing heat. If you were already pushing your subs to the point where they got pretty darn warm and it just hits summer, there's a good chance if your subs reaching thermal limits even faster since the air is not as cool or sometimes hot and overall cooling decreases.
Mechanical limits:
Playing too low with a lot of power or not good enough of a subsonic filter and too much port area or too big of a box can cause the sub to unload or worse, bottom out which the coil formor slams against the bottom of the magnet or worse, it jumps out of the coil gap. However its pretty rare to get to that gap jumping point. Poorly designed subs with different parts as in too strong of a motor and too weak of soft parts can cause the sub to reach mechanical limits a lot faster.