OK, now this is getting weirder!

I've just clicked on several sort of random YouTube videos that are not my own uploads or "favorites" and some of them play fine but others don't (just the black box with no "play" button.)    There doesn't seem to be any discernible pattern to this.

This one plays fine:
Nathan Moore - "Understand Under"

This one didn't:

And NOW I tried this one
http://www.youtube.com/watch?v=aHw3Y36igW0&feature=relmfu
and the still black box had a note saying "Plug-in Failure" and I got a Problem Report for Flash Player, saying Flash Player had quit unexpectedly.   So it sounds like it may be an erratic failure of Flash.   I'm resetting Safari first to see if that helps.