So what you are saying is that the system can stay on properly for progressively shorter periods of time after the initial failure. That indicates a heat dissipation problem. What follows is basically some suggestions on how to confirm that hypothesis and, if it is correct, attempt to rectify the problem.
My first suspect, since the computer has trouble turning on, would be the power supply itself. CPU issues can also cause such troubles but will normally be an either-or situation (the computer either works, or does not work at all) or are such that they don't prevent the computer from powering on.
First check to make sure that the power supply fan (it almost certainly has one) is clear of debris, is moving freely when you turn the computer on, and the PSU air intake isn't clogged by dust or otherwise blocked. If it's full of dust, remove the power supply from the computer, unplug everything from it, place it on your desk and let it sit for a few minutes before carefully using a vacuum cleaner on low to medium power to remove the dust. Do not blow into the PSU; you risk spitting into it. Letting it sit for a few minutes allows the capacitors to fully discharge and is probably a little overkill, but better safe than electrically shocked. There are some pretty serious voltages and wattages inside even a small power supply.
If that doesn't help, I'd try another power supply, to rule out PSU issues. Make sure it can provide the amount of power and current needed by your system. If you are unsure, overprovision rather than underprovision; underpowered PSUs can cause all kinds of hard-to-track-down issues.
Because one indication is that the screen turns off, there's also the possibility that there's something wrong with the graphics card or its settings. Is the graphics card actively or passively cooled, and if actively cooled (a fan on it), is that fan working properly? Check to make sure the graphics card is seated properly and that the slot and contacts are clean.
Try booting off some live CD or OS install media and just let it sit there for an hour or two not doing anything. If the system stays on then, it means that there's some sort of issue with your installed OS or software, and not the hardware itself. (Also note that most such media, when booted, will be running the CPU at constant full throttle. If it's a CPU heat dissipation problem, that will exacerbate the situation, so if anything, it should fail sooner than normally.)
Use some tool that allows you to monitor the CPU core temperature while the system is running normally, and keep an eye on that reading while you don't do much with the computer. Don't worry about the CPU getting pretty warm as long as it stabilizes at some value. Move the mouse occasionally to make sure the computer is still responding. If the CPU core temperature just keeps rising, it might be a heat dissipation problem for the CPU. (Modern CPUs have safeguards that shut them down if the core temperature starts getting much too high, which appears to the user as the computer either simply freezing and not responding to anything, or shutting down completely seemingly randomly and without warning.) Check the CPU fan, heat sink and if both of those check out, apply new thermal grease between the heat sink and the CPU chip.
If all else fails, see if you can try another motherboard. Keep in mind that with ATX, the power button simply controls power state transitions, which the motherboard reacts to by turning on or off various components. The only way to fully turn off an ATX PC is to use the power switch on the back of the power supply (usually located next to the AC mains cable receptable) or unplugging the AC mains cable.