2 types of distortion: 1) harmonic distortion, 2) improper driver movement distortion (fizzing out basically).
1) is what that causes boominess / rumbliness in certain headphones with bass, that stands out from what headphones with very tight controlled neutral bass sound like. Basically when a 50Hz tone is played on headphones with high bass total harmonic distortion, you're hearing half 50Hz bass, half "junk" (bass of other frequencies in different harmonics), this causes the boom/rumble, aka dirty looseness to the quality.
2) is when the driver simply cannot physically handle the piston action of very strong bass analog signals and peters out, so instead of bass, you hear something hollow and off.
There is also a 3) which doesn't really count, is when the amp clips, but that is not a flaw with the headphone but just ****ty signal because the amp doesn't have enough juice.