When Hollywood asked Verizon and AT&T to monitor and block customer trasmission of suspected pirated content, Verizon said no. AT&T is apparently looking into it. See: Verizon Rejects Hollywood’s Call to Aid Piracy Fight - Bits - Technology - New York Times Blog.
In somewhat related news, the European Court has said that ISPs can't hand over customer data to rights owners seeking civil remedies, unless local laws explicitly allow it. See: E.U. Court: Downloaders Can Stay Private - New York Times.