YouTube video Stops

Try uninstalling Adobe Flash and reinstall the Flash 10.1 beta. Also, reset the router or modem and make sure than you're not maxing out either your the routing table with torrents or your bandwidth.

I would suggest you to try installing Google Chrome, but it could be useful to know what Operating System you're running (Windows Xp/Vista/7, Gnu/Linux of what flavour, MacOS 9.2/X, etc ?)

Best regards

EDIT: while you're waiting for some REAL solution, you could be interested in this: http://www.kissyoutube.com/ When you are watching a video with the behaviour you've reported in your answer, add "kiss" to the URL