A personal anecdote:

I had several roommates, and we each were responsible for a utility. I was responsible for internet, and Cox was our provider.

I received multiple e-mails from Cox about copyright infringement. I can't recall them, but I remember it being serious enough for me to tell people to stop.

Thinking back, I feel like Cox's position is right and fair; let users know they're being observed by copyright holders, and inform the user that they could be compelled to provide their identity to complainants.

But ultimately, the responsibility to "stop" the supposed infringement is on the holder, not Cox.