I assume you mean the old 3rd generation and not the 4k 3rd generation? Perhaps try connecting it with an ethernet cable, just for testing at least, you could move it closer to the router if you don't have a long cable. If it doesn't stop playing with a hard-wired connection, it's probably a wifi issue.
Also, could this be caused by the remote somehow? When was the last time you changed the battery? I had two Apple TV 3's and replaced them with Apple TV HD's but continue using the old aluminum remotes. However I have replaced both remotes because they started getting flakey (buttons not responding). I wouldn't think a bad remote would cause it to stop playing, but who knows? I suppose you could remove the battery after starting a movie and see if it still quits.