What evidence do you offer to prove that it wasn’t you? What evidence does the prosecutor or civil party offer to prove that it was you. Yes, you can be held liable if the trier of fact (jury or judge if it is a court trial) believes it was you. Otherwise, not.
Civil suis are determined on preonderance of evidence. Who’s case looks stronger?
The party suing will have a certified record saying their investigator downloaded a song from your IP address; they have the equipment to verify that was the correct IP. They may get a discovery process (not sure how this works) to examine your hard drive. Failure to cooperate will go against you.
In the case of Jammie Thomas, having the local computer store replace her hard drive between the notice letter and the discovery process, also being evasive or misleading on many other topics, might give the court reason to doubt her word.
You don’t win civil suits by being a difficult smart-aleck. The judge has probably seen every idiot in the book, and dealt with them as a lawyer before becoming a judge. Just because your story sounds cleevr to you does not mean anyone else will buy it. OTOH, judges are often not too smart about this sort of tech either, and often court-building office politics may have as much to do with the result as facts do.
If you’re like the old couple in “Pump Up The Volume” whose phone line is being hijacked by someone else using radio - well, sucks to be you in court. “I haven’t a clue, couldn’t be me” is not a great defence. This is why it’s always a good idea not to escalate a fight when it means legal fees will result.
Then there’s the “Oscar Wilde” principle - suing when you are wrong and will probably lose is not a good idea.