This is just a private company threatening to sue you (for more) if you don’t pay. It’s extortion. As chrisburton said early on, nothing comes of these threats in most cases. But similar lawsuits have been brought; and a few have won, so there is a risk.
On a side note, I’m sure you’re all aware that “downloading” something doesn’t consist of literally picking up the file and moving it. You get a copy. Because of how the internet works (your packets are routed over a number of servers, not necessarily apparent to you), it may actually be copied anywhere from once to dozens of times. RIAA have actually argued that every single one of those “ephemeral copies” are illegal, and that they have the right to collect damages from each one. It’s absurd. It’s like shoplifting a $1 pack of gum and having to pay $1M in fines because, well, you took a lot of steps after you left the store.