Considering that cutting swords remained popular for nearly all of the chainmail life-span, I doubt they just bounced off
Kinetic energy hurts like hell, even with mail on and some swords were still very "choppy" in this time period, even without parallel edges and very wide fullers... Sure, an axe or a mace will be a small margin more effective, but overall, wearing armour reduces damage significantly; it doesn't negate it entirely.
Now, granted, the stuff depicted in most period art pieces is a gross exageration: I do not believe that, however poor the quality of a mail, a sword would be able to slice it open and amputate an arm or split someone skull open through a mail coif or run someone through with a thrust. But even so, poor iron quality could be a factor too.