I have owned a 23″ Cinema Display for the past one and a half years and I really enjoy using it. It recently stopped working and would not wake up from sleep. The problem turned out to be the power supply brick which I just replaced for $80 (including shipping).
I did suspect the power supply and took it in to a repair shop to see if it was the problem or not. It worked fine in the store. However, we did not leave it plugged in for long enough to realize that it would shut down after a minute from a cold start and after a few seconds on subsequent tries.
So, if you have a problem with your Cinema Display, verify the power supply before sending in the whole unit for repairs (~ $400).