Since we moved, our Apple TV has been giving us trouble: we would rent a movie on the Apple TV, and instead of playing it, the Apple TV would just display
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Rebooting, then trying again, would sometimes solve the issue. Trying to play a trailer before trying to play the movie seemed to increase chances of the Apple TV actually downloading the movie, too.
Finally, some forum or other pointed me to the correct solution: For some reason, the Apple TV did not play nice with our ISP DNS server. Simply changing the DNS server in the Apple TV settings to one of the Google DNS servers (18.104.22.168 and 22.214.171.124) solved the issue. Any other DNS will probably work, too.