Applecare says it is a problem with the YouTube servers. I have the
same problem on my iTouch for days at a time. It comes and goes with
no changes to the iTouch. Connectivity isn't the problem because I
can see the listings, pull up the videos and read the text. I search
for videos or pick ones at random and none of them work. Going to
youtube.com with Safari yields the same result. All I get is "This
movie can not be played."
This is really maddening in that no one from youtube will comment on
it at all. You can not call them, period. They do not even have an
email support option. So, you are just stuck! They obviously do not
care about this problem. A quick Google search (the irony...) lists
tons of places where people are asking the same question as you. But
no body from youtube will lift a finger on it. Well, I guess they are
showing us at least one finger...
The only thing you can do is try again latter and MAYBE it will be
over.