Well subs are nothing more than a large speaker driver...so to answer your question yes they can distort.
Anytime you are ending up with a different sound wave from what is on the original media...the sound has been distorted.
And no, you can't just turn subs up as much as you want...unless you are really good and then you're just pushing the limits until your subs blow out... lol
Oh...and since I see this question possibly popping up later in the thread I'll answer it now...
Q: How much power can I throw into (insert sub name) before it'll blow?
A: Power DOES NOT blow speakers...DISTORTION is what blows speakers. The only reason that people have been thinking that power blows speakers is because you have a smaller margin of error before the speaker gives out when you are working with a larger amount of power.
Hope that helps!