Having direct knowledge (as opposed to merely having reason to believe) that copyright infringement is occurring.
Some copyright laws require web hosts to remove content from their servers if they posses "actual knowledge" that the content infringes copyright. Under such laws the hosts may become liable if they do not remove the content.
For example, Section 512(c) of the U.S. Digital Millenium Copyright Act reads:
“(c) Information Residing on Systems or Networks At Direction of Users. (1) In general. A service provider shall not be liable for monetary relief, or, except as provided in subsection (j), for injunctive or other equitable relief, for infringement of copyright by reason of the storage at the direction of a user of material that resides on a system or network controlled or operated by or for the service provider, if the service provider (A) (i) does not have actual knowledge that the material or an activity using the material on the system or network is infringing;