I keep getting a message that "Apple TV can't connect to the server at this time. Please try later.

Try the following steps, check whether things are working after each step where appropriate, before trying the next.

1. Restart the Apple TV (Settings > General > Restart).
2. Restart the Apple TV by removing ALL the cables for 30 seconds.
3. Restart your router. (Also try removing it’s power cord for at least 30 seconds)
4. Reset the Apple TV (Settings > General > Reset > Reset all settings)
5. Restore the Apple TV (Settings > General > Reset > Restore)