Sam Cook's ISP informed him that he has been downloading illegal material. He has done no such thing. He asked me to help him solve this problem.
Neither your ISP nor anyone else can actually tell what you are doing on the Internet. But they can follow the activity of your public IP address--the one your router uses to access the Internet. And if someone else uses that address for unsavory purposes, you could become a prime suspect.
As I've previously explained, your public IP address is readily accessible on the Internet. Anyone can use it to discover your general location (your neighborhood, not your house) and your ISP. Your ISP can identify it as yours, and will do so if subpoenaed.
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So how could someone else be using your public IP address?
It could very likely be someone you live with. Everyone using your home network is using the same public IP address. Have a very serious talk with your family or roommates.
Another possibility: A neighbor may have hacked into your Wi-Fi network, and is now using it without your permission. You can stop this by changing your Wi-Fi network's password and tightening your router's security.
Something else to keep in mind: Most home Internet connections use a dynamic IP address. From time to time, your address will change. When your ISP accuses you of illegal activity, ask them how long you've had that particular address. Yes, they should have looked that up before contacting you, but we all know that ISPs often lack sterling customer service records.
Finally, it's possible that someone far away could actually be spoofing your IP address. Discuss the possibility with your ISP, and ask them to give you a new IP address. If they comply, and the illegal activity stops, you've solved the problem.