Most websites are more likely to violate the "old" copyright law
than they are to violate the Digital Millennium Copyright Act, but
the prime way sites find themselves up against DMCA is by
distributing software designed to circumvent DRM and similar
copyright protections.
DMCA takedown notices are often sent in response to posting
infringing material, but that's a violation of the plain old
copyright law; DMCA just makes it easier for rightsholders to
respond.