I got a notice saying I downloaded something I didn't. My roommate said he didn't either (yeah...). My notice came from a lawyer who opened something up via a court. He would call me and leave multiple voicemails a week saying he was going to "take me to court soon" if I did not give him $3k in settlement money. This went on for almost two years. I kept ignoring his calls and eventually he closed the case. There were 30 other people lopped on this "bit torrent sharing" case and half of them paid up, so I guess he moved on.
Anyways, I've done a lot of reading on this type of stuff. What type of notice did they get? Sometimes you'll get a warning. Sometimes it'll be a company just trying to get a small fine of $100. In my case it was a notice from my ISP saying my personal information was being subpoenad.
If it's from a porn company they are most likely looking for easy targets to extort money out of and they dont want to actually take anyone to court.
That was a scam that was happening a lot like 7 years ago. They knew they couldn't win the cases, but did have proof of your downloading. So, they would send you letters trying to trick you into settling for a small sum. Usually 3k as you said. A lot of people would agree to pay because they knew they were guilty. The companies, however, knew they wouldn't win, and even if they did, they would spend more fighting it than it would net them. Thus, they would drop them if it actually was not agreed upon by the customer to settle. I think someone even filed a suit calling the scam out in the past. I can't remember.
The only case I ever remember going to court over this was a poor woman got tried over this for downloading cartoons for her kids illegally. The backlash the company got was so incredibly immense that they basically found it as a total negative. The lady couldn't pay the sums anyways that were ordered, and the company got bad press for going after a poor family just trying to let their kids watch cartoon movies.
They have said the best response to these is to not acknowledge them at all. Acknowledging them shows you are aware of them. Responding to them is also showing you are aware no matter what your response is, making a subsequent one as knowingly doing a wrong practice. And, of course, responding that you did is an admittal of guilt. If you see a copyright notice, do not open the emails, read them, respond to them, or take any action in regards to them.