Well I'm not sure if this is the case in apple monitors but in most lcd monitors the first component to go bad is the capacitors in the power supply. This will cause the display to turn on very dim and take several minutes to brighten up or to display black. If you think the monitor is at fault and you have checked your connections thoroughly it might be worth taking it apart to look at the capacitors. If they are bad they will sometimes have a bulge on the top, they should (when working correctly) be perfectly flat. They can go bad without the bulge but this could be a good place to start.