I've had a brand new 32" Full 1080p TV for a few months now, and Have recently upgraded my PC, my graphics card is an Nvidia GTX 460, the PC and TV are connected with a DVI to HDMI lead, the desktop has no problems running at 1080, runs like a charm, I even managed to get Starcraft 2 to run at 1080, nothing else though.
Here's how it goes down:
Start a game, set resolution to 1080.
Screen goes black, sounds of game still in the background.
Game menu flashes for all of a second, then black, again the screen flashes back a couple of times.
then nothing. Screen shows "No signal from device!"
Then it resets back to 600x400 or 800x600 or whatever it started at.
And this isn't just 1080, it's any other resolution other than the one it's already set at.
That is indeed a problem. What PSU? Install all the latest driver to the video card. I do not know if installing driver will help, but better give it a try. Make sure that your card is all the way plugged in and so is the video output.
Sounds like you are having a "handshaking" issue between the monitor and your vid card. I'd try swapping vid cable and dvi-hdmi adapter. If no change, then I'd try the PC-HDMI on another 1080p monitor to see if I could localize the problem to the vid card.