Yes, but that's what the safe harbor clause is for. If a user uploads something, the site owner is not liable provided they comply quickly with takedown requests. It's why YouTube can allow users to upload videos, and a similar principle is the reason 4chan is still in existence despite the hundreds of thousands of child porn pics that have been uploaded to it.
[0]: http://news.cnet.com/8301-31001_3-57327815-261/lawsuit-claim...