http://www.charlierose.com/watch/60268556
15':25" into this clip (the one titled 'The Federal Reserve'), Charlie Rose said 'you know better than I,' and then quickly said again: '... better than me.'
He seems to not as much correct himself as attempt to cover both sides of the fence.
It is remarkable that someone like him, grand master of the language if there is one, conversing with assorted more or less competent practitioners, still feels uncertain, and feels enough urge to even take a second to switch sides. Remarkable, for nothing, but remarkable!
Worst part of this is that he was right the first time, and "corrected" himself with an error.
Quick refresher on "I/me" in comparisons:
She likes John better than I. - means that I don't care much for John, but she has a higher opinion of him.
She likes John better than me. - means that if she had to choose between John and me, I'm getting left out in the cold.
In other words:
She likes John better than I (do).
She likes John better than (she likes) me.
Edit:
Actually, I suppose it's She likes John better than I (like John).
but:
You know better than I (know).
You know (what?) better than (you know) me (?).
I can't make sense of it using me.
So, Charlie seemed to have momentarily decided that proper grammar was not a good thing in real life. He wanted to show sympathy with opposing sides. But he really should not cater that way, not with grammar.
I know nothing about Charlie, but it sounds as if he would have been better off making a choice, one way or the other, and sticking with it.