Depends, largely on how motivated people are to identify you.
If you infrequently torrent a few games and movies, using a VPN will probably be enough to deter most of the law firms specializing in churning out copyright notices. They go for the low hanging fruit first, aka the users without a VPN.
If you on the other hand were to engage in torrenting highly illegal stuff, and were therefore to become a target of state law enforcement, you'd be in a whole different kind of trouble. A VPN would likely not be sufficient.