It turns out that I was running a dev release of PHP 5.6 from a PPA and not the official Ubuntu release. With Brad Bell's help (thank you so much!) I got on the right track to looking into PHP and realized that that was the problem. After purging the PPA and reverting to PHP 5.5.9 on the official Ubuntu repo, the problem seems to have gone away. Hopefully never to return again.

Thanks again, Brad!