At least a few years ago in Sweden, all cases where someone had payed a fine for downloading pirated material due to such letters was because the person admitted guilt. The anti-piracy organizations had no way of forcing someone to pay, because none of the evidence would hold up in court. They tried to submit screenshots of IP addresses, but since it's easy to spoof it wasn't enough to convict someone.

I don't know if this has changed in recent years, as I haven't heard anything about it recently.