RIAA asks for around $3/song when they catch someone pirating a large number of songs. The small handful of people who ended up higher got that way by being idiots.
E.g., the most famous case, Jammie Thomas-Rasset, was caught sharing 1700 songs. The RIAA asked for $5000.
She refused, so they sued. They sued over 24 of the songs. Why not sue over all 1700? Probably a couple reasons. First, if they sued over 1700 and won, the minimum damages they would receive would be $1 275 000 (statutory damages of $750 [1] per infringed work x 1700 works). Only suing over 24 cuts the minimum to $18 000. They were willing to settle for $5 000, but having to go to trial adds some expenses and so $18 000 is pretty reasonable.
Second, for each song they do have to file some paperwork proving that they have standing to sue on behalf of the copyright owner. Every bit of paperwork they have to file is time and money, and every bit of paperwork offers the defense something to challenge to further take their time and increase their cost.
Any non-moron looking at the above math would beg to accept that $5 000 settlement as soon as they found out that they were in fact being sued. But not Thomas-Rasset. Worse for her, she was a very unsympathetic defendant. She tried to pin the blame on her kids, she tried to destroy evidence, and she lied about that under oath and was caught at it.
The jury gets to pick the damages and instead of going for the low end of statutory damages they went for something higher up, and she got hit with a large fine (around $200k).
And then what did the RIAA do? They offered again to settle for some reasonably low amount. She refused, got another trial, lost again (because of course nothing had changed) and that jury went high, going an order of magnitude higher than the first jury. The RIAA again offered to settle, although now they were asking $25.
Eventually she got an appeals court to lower the damages back down to around the first trial's amount. Again the RIAA offered a much lower settlement, if she would agree to make a video about copyright infringement. She refused.
If you are going to pirate, fine, but be an honest and honorable pirate. If you get caught and are offered a reasonable settlement (and 3x what it would have cost you to buy the works without pirating them is reasonable [2]) take it. Don't waste the court's time with going to a trial you have pretty much no change of winning and that when you lose the minimal damages will be more than the settlement offer.
[1] It can go down to something like $200 or $250 per work if the alleged infringer can prove they were an "innocent infringer", which means that they did not know the works were copyrighted and had no reason to know this. That would be very hard to prove.
[2] It has to be more than the retail cost because if the worst that happens to you when you are caught is that you pay what it would have cost to buy retail, there is no incentive not to pirate.