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The problem definately sounds power supply related, but since you have replaced the PSU, I would start to suspect the motherboard, CPU, or RAM.
Just to make sure it has nothing to do with the power or power supply, try moving the system to a different location in the house/apartment/whatever, turn the switch on the back of the power supply off for a minute or so, plug the machine in, turn the switch on the power supply back on, wait a few moments, and then try to power the machine on. Let us know if that has any affect on the system.
Have you tried running the system with only the bare minimum of components? Try removing everything from the system except the motherboard, CPU, heatsink on the CPU, and RAM. These are the bare necessities to allow the machine to operate. Make sure that the fan on the heatsink is still connected to power. Try to boot the machine when it is in this state. Let us know if it runs longer like this.