Blackmail, pure and simple.
You contest it if you're alleged to have downloaded "Iron Man 2", and if your name shows up in local newspapers so what, you know it'll cost the more in lawyers fees than they'll ever get awarded in court.
But if they claim you downloaded "Teen Anal Nightmare 2", they are pinning their hopes that you'll not want the world to know what you knock one out to, so you'll pay to settle out of court.
That was the strategy of ACS:Law in the UK and they just got their ass handed to them by a mass denial of service attack which caused them to expose the root of their network to the world, including all email correspondence for the last year. That breach of data protection laws will likely bankrupt them and has seen the, ahem, gentlmen porn producers and senior lawyers involved become the target of spam, ID theft, harrasment and a abrupt halt to their dubious quasi legal blackmailing. The irony is that those individuals named in those email exchanges can now sue the porn producerss, the law firms and their own ISP companies for releasing their data in an unprotected format.
It's going to be costly. How secure do you think "West Coast Productions" & "Adult Copyright Company" computer systems are? Not secure enough I'd guess, for a dedicated team of robin hood hackers who want to put a stop to this blackmail and copyright witchhunt.