Extreme Buffering after 10-20 Minutes of normal playback

This thread seems to be tracking a very similar issue regarding the AppleTV client.

A plex dev has already commented on a potential cause, and a limited group of testers seem to have had success with a test patch.

Until the next AppleTV build gets pushed, either turn off the enhanced player, or roll back to the AppleTV 2.15 plex client if you can.