Several months ago Google launched a new search algorithm aimed at downgrading the ranking of those who have many DMCA requests that were legit.
Since then, we've seen the number of DMCA filings spike. Was it related to the algorithm update or more savvy publishers trying to claim their content - I don't know.
What I do know is that even several months later, there are some webmasters still nervous about this update.
One WebmasterWorld thread has a webmaster asking if he should worry about receiving "8 DMCA complaints in 10 days time." The quick answer is no. You'd probably need a heck of a lot more than 8 DMCA requests for Google's algorithm to kick in.
But any user generated content site should have policies and measures and checks to ensure unique, valuable and not-stolen content is used on the site. If you do not do that, you can be hit by not just the DMCA algorithm but algorithms like Panda.
Forum discussion at WebmasterWorld.