Yes, that happens. Some hosts will be more strict about it then others. We require an actual letter from the offending party before we forward to the site in question. A lot of DMCAs seem to be faked.
Actually, you should be very proud to receive some massages from DMCA. Usually, they are not so nice, but massages is always good... Matter of fact, I have some discomfort in my shoulder, I'll send them an email right now...Who knows, maybe I get a massage as well...