Twitter has registered itself with an agent for the reciept of DMCA notices. This gives it the power to remove any infringing substance on twitter or a third party server on request or their own belief.
Recently, one Jennifer Rodinelli Rieilly filed a suit in California federal court for not having met her several requests to Twitter for removal of about 65 posts infringing her copyrights. On November 18, 2015, Rielly sent 28 notices to Twitter.
The material still remained on twitter during filing the suit.