Google stops plenty of companies from shooting themselves in the footPlenty of companies have taken advantage of the tools Google provides for them to request the removal of infringing content from search results. In fact, Google sees millions of these requests every week.
It goes through all of these requests and doesn't approve the obviously wrong ones, but many erroneous ones do get through.
Most of the time, those errors affect legitimate users, but many times they affect the copyright holders themselves. Plenty have asked Google to block videos or music they themselves had published.
The latest in the long list of companies going by the "shoot first, ask questions later" mantra is HBO which has been asking Google to remove from its search results links to HBO.com as well as to several other legitimate sources.
This is nothing new, plenty of companies have made the same mistakes, you can't send out hundreds of thousands of requests each day and expect all of them to point to legitimately infringing content.
Luckily for the likes of HBO, Google does actually review these requests and catches most of these errors before these companies manage to successfully shoot themselves in the foot.