Unfortunately, the problems with streaming more than one video or song without having to reset itunes that were prevalent in 5.1 are still there. I had really expected this to have been fixed. Looks as if Apple has no intention of correcting its mistakes. It is this attitude that will be the ruin of the company!
ATV 6.1 Streaming still not fixed
Don't really think it is local. Here is a link to others with the same issue....
So let's diagnose it, because even on this old as sin setup I have, it works perfectly fine. What are you serving up, how are you serving it up, and to what is it being served?