Except the law is not on their side. This is from an article about this on Wired.com, though, so you're welcome to take it from whence it comes.
the basic gist of this is that in the UK, where these guys have been practicing, there is no statutory claim to damages, and the lawyers in the UK system in a case like this would usually be able to claim only as much as the retail price of one item in damages. That would mean 75p in the case of a single downloaded music track.
The law firms are sending letters of demand for much more than this, and sending them to people in financial difficulty - who cannot afford to get legal representation, and who often pay up to make it go away. Hearing about massive damages awarded in cases similar to this in the States probably is a factor.
The lawyers typically don't go after people who haven't paid, and bring them to court. But one of them is considering moving from the UK to the US just because of the statuary damages angle that RIAA have managed to make law.
The wired article is here -http://www.wired.com/epicenter/2010/10/the-legal-blackmail-business/ - so everyone can ignore that one, as well, and write whatever comments they feel.