Apple TV 3 Crashing Using Home Sharing

I'm trying to play a 3.49 GB (2 hour 45 minute) video stored on my iMac, via Home Sharing to my third-generation Apple TV. It works, for a minute, then the Apple TV crashes and reloads the main menu. So I try playing the video again. It plays for a minute, then the Apple TV crashes and reloads the main menu. And so on.

The problem clearly lies with the Apple TV. I just tried playing exactly the same video, or exactly the same Wi-Fi network, via Home Sharing to an iPad mini, and it played perfectly. No crashing.

So, why is the Apple TV crashing? Does anyone know? It's really frustrating because I can show the file just fine on an iPad mini, but not on the TV (where I want it).