1 week ago a Wikiki Honolulu Apple Store genius told me she knows ATV3s loose connection; that Apple does not allow her to tell me exactly what the problem is; and she is not allowed to tell me how to fix it myself; that I must bring the device to a genius who will check it and fix or replace it !!!! Because I was on holiday and did not have the ATV3 with me, her offer was not helpful.
I have no reason to change my previous post, that I repeat here....
Scan the Support Community sites and you will find: Apple TV losing connection is prevalent. The many postings point out the problem occurs with:-
1 Apple TV1,2and 3;
2 with different brands of router including Airport Extreme;
3 with current and past versions of iTunes;
4 after varying periods of working well.....minutes, hours, days;
5 losing connection with Home Sharing and losing Airplay;
6 whether the router and Apple TV are in same room or separated by distance;
7 after trying innumerable posted 'fixes';
8 after exchanging for a new Apple TV;
9 with all the recent operating systems -Mountain Lion, Lion, Snow Leopard, Leopard and Lion.
10 when it loses wifi, the ATV will work immediately if you plug in the ethernet cable.