I assume you've tried unplugging the Apple box for a short bit, and then restart it, and try again.
You could also exchange the box for a new one.
Switch out the Hdmi cable, switch boxes between the two TV's.
Does this behavior happen on every app?
You seem pretty positive it isn't your network in any way, but rebooting your modem/router etc., can't hurt. Or if you can use the ethernet port try that.