The Pew Hispanic Center, a research group that studies issues, attitudes and trends among the Hispanic population, estimated in 2006 that almost half of the 10.8 million illegal immigrants in the U.S. came here with visas and stayed after they expired.
They don’t even need to overstay. I’ve known many people who violate the terms of their tourist visa by working (typically cash only) but still flying out of the US for a few days before their visa expiration and then flying back in with a renewed visa.