I'm having similar issues playing videos on iOS. First things first, I'm on a iPad 2, using iOS 7 and a 8Mbps connection. I primarily use safari & I'm geographically very far away (SE Asia). Videos for me always start at low quality, then in 30 seconds upgrade to medium & another 30 seconds later to high. After that I don't have a problem till the last 5-7 minutes of the video, where the video either stops or drops to low quality. I have no problems with YouTube content of any resolution or for that matter any videos anywhere.
Additionally, if I try to start a video on the high setting, it just won't start and give me an error 'the operation could not be completed'. Also, if I might add, all these problems disappear when I use a PC, navigate to GB.com on my xbox and play videos on whatever version of IE it uses (lol!! Wat?!?!).
One last thing..I worked with my ISP to optimize my connection. It seems data transfers through ISP's can fluctuate dramatically when trying to stream anything. If it drops too low, it will downgrade the video quality or stop altogether. It used to happen a lot before said optimization..now at least I can watch 95% of a video in high quality, till it drops/stops in the end.