My new Apple TV (4th gen) had been performing fine, until this evening when a movie I just purchased from iTunes started stuttering/pausing at random spots while streaming from my local iMac's iTunes library. Stopping/pausing/replaying portions of the movie showed that the problem wasn't a particular point in the movie, since the stuttering occurred at different points in the movie. Replaying previously stuttering points play fine, only to have the stuttering occur a bit later in a different location in the film. I'm running the latest retail build of MacOS X 10.11.1 on my iMac, and the latest build of iTunes, as well.
My Apple TV (4th gen) is connected via Ethernet (gigabit) to my local area network (LAN), and my network router is an Apple AirPort Extreme (802.11 ac). Everything else on the network is performing normally. I checked my ISP connection with Comcast, and it is also performing normally (30mbps).
Two movies I recently downloaded are affected, Ant-Man and Inside Out. Last night I downloaded Mission Impossible - Rogue Nation, and it ran smoothly, without incident. Tonight, after the stuttering problem with the two movies I mentioned, I watched another movie I recently downloaded, Star Trek 4, and it played fine. So, I'm a bit perplexed.
I tried switching the Apple TV (4th gen) to WiFi (ac) and the stuttering persisted, so it wasn't the network connection. I then copied the movie to another newer model iMac I have running the latest build of MacOS X 10.11.2 (beta) and AirPlayed the movie to the Apple TV (4th gen), but even the copy stuttered on the new Apple TV over the AirPlay connection. I then played the source movie from my primary iMac over WiFi to another Apple TV (3rd gen), and it played normally.
From these steps I'm left with the impression that something in the encoding of the two problematic movies doesn't like either the version of tvOS or the new hardware running on the new Apple TV, since at least one of the problematic movies this evening played fine on my old Apple TV.