The netgear router is only wireless G but this has been spuperceded by wireless N which offers faster speeds and greater range. Apple TV supports wireless N so a new router could help. Have you tried an ethernet connection from the router to the box just to see if you can get a satisfactory result, after all your bb is 6mb so that will be the limiting factor, not, ultimately, the wireless connection speed. Bear in mind that if you are sharing your bb with other computers etc that you won't get the full 6mb connection to the apple TV and the streaming might therefore keep pausing.
Sony Xperia Z5 review: Hands-on with the phone which the Z3+ should have been