I used to have a similiar problem with the computer I'm using now..
Basically, it was the motherboard not cooperating with the video card and I needed to replace the board.
Do the suggestion Seti made first.. but going into sleep mode from an O/S would mean the O/S needs to initialize which would not occur once you turn the computer on.
This means, that it seems be a problem related somewhere as far as hardware.. because the monitor is not turning on due to the lack of a signal coming from the video card. If you have onboard video, and your not using it, or a really cheap PCI video card.. try it out.
The problem that I experienced was that I would turn on my computer and everything would seem fine. Luckily for me, I could see clearly that the video card wasn't booting (my video card has a fan, and blue LED lights, and neither were on). Well, I went through 3 RMAs with BFG Technologies and finally realized it had to have been my computer.. something internally.
Well, let me say this to.. I only had the problem when I would turn my computer off for say 5-10 minutes or more. If it decided to boot successfully, I could leave it on and have no problem.. but once I turned off the computer and went to boot it, the problem occured. I even brought it to a technician who at the time I considered to have more knowledge than I did.. who wrongfully told me the card was dead, when it wasn't.
I had an MSI NForce 4 Premium Board I think it was.. with an nVidia 6800GT.
Since I changed my board to an ASUS A89-SLI Premium board, I've had no problems at all.. except the need to update my chipset drivers (but thats a different story).
Needless to say, do the following:
1) Check the connections.
2) Try another card, if you have one availible.
If that doesn't work, I'd bring it to a technician and ask them to find the exact source of the problem. They can test the video cards and other components individually.