If you've tried two different adapters, it's probably an issue with the display.
It's possible. What happens when you jiggle the connection? In any case, it's tough to get parts and a crap shoot akin to a shade tree mechanic otherwise. See here for a parts listing: http://oldsite.auburnschl.edu/Technology/protected/applemanuals/cinema_hd_displa y_23.pdf
I just tossed my old 23" Cinema HD Display a couple of days ago because it had an intermittent heat related problem. It was either the inverter board or the main board. It just wasn't worth messing with it.