In my previous post I noted that this problem was caused in my case by some of the USBPWR jumpers that select between 5V & 5VSB were missing. This suggests that ASUS are testing the 5V Voltage (not the current) is within range. Hence the problem is likely related to the PSU and how much power is being drawn from it.
So I suggest the following likely problems (when the jumpers are not missing).
1) Too many things plugged in to USB ports (obvious really). Unplug all USB devices and try again. If it works one or more device is faulty or you have too many drawing power from one USB port. Its possible they might all work if you move the devices around!
2) Another faulty component that is causing the 5V line to drop - disconnect everything and try with just your video card and one memory DIMM. Preferably if available try replacement video or memory. In particular if you have a half decent 3d video card try a cheap old PCI card as these will draw far fewer amps from your PSU (see 3b).
3) PSU problem either
a) Faulty
b) Not powerful enough for the system - which in a new system usually means its underrated - in older systems they are "tired" and/or the system has built up extra resistance over time and therefore presents a greater load.
It's possible in case b) that you might be able to use the USBPWR jumpers to switch some of the USB ports to use the 5VSB power line instead of the 5V power line (or less likely vice versa). This might be enough to put the voltage back into the acceptable range and the system might work - however your system might not be all that stable if your PSU is this close to its maximum performance.
You could check the voltage with a volt meter if you have one and know how to use it. You can use this
http://en.wikipedia.org/wiki/ATX to see where you should test.
If you need a better PSU check the current ratings that are on the side of your current PSU and buy something with higher values particularly for the 5V line. Remember that the values are manufacturer figures - cheap ones are often vastly overstated.
4) A motherboard fault presumably affecting the 5V power.
5) A power short to earth - unlikely as I don't think it would start up your screen in this case.
6) A combination of the above.
If you have a significantly better PSU available and all the USBPWR jumpers are in place a simple first test would be to swap the PSU and give it a go.