I just recently faced this problem, while I have high quality broadband bandwidth and each devices (laptop, apple-tv ,etc) can stream 1080 flawlessly, but the moment I hit the apple tv button, the online streaming (fb or youtube) stops playing back. If I cancel sharing the screen with apple-tv it resumes playing back!!
p.S: I have this problem only with online streaming, offline video playback is fine on apple-tv. Bandwidth test with apple-tv is passed without any problem.
laptop: mac-Pro 15" retina
apple-tv model: A1469
apple-tv software: 7.0.2(6915)