I am interested in this because I have apparently the same problem.
Power gets to PC, fan starts, USB is powered, power light, a flash of HD activity, tries CD drives, but no progress, just a black screen, with no disk activity other than the initial. There are no beeps.
I need to try to start up 10's of times - the number varies - before it will power up. Have you tried many times to turn your PC on ?
I have made various changes to the PC shortly (some months) before this problem, but the problem didn't emerge immediately - some months later. I think it's got worse over the last 7-8 months.
Possible 'quick' solutions, which you might consider (not all of which I have tried):
Power supply - defective or under powered.
Fan problem (have tried replacing the fan though)
Cables not seated properly (I have tried reseating them though)
Missing Northbridge heatsink
Short underneath motherboard (new idea) - possibly caused by 'lost' Northbridge heatsink anchor point (a loop of metal)
Defective Capacitors on motherboard (a difficult job, but it can be done)
Compaq Presario S5160UK DT261A under XP/SP3
Processor - Celeron 2.7 GHz
Motherboard - MSI MS-6577 v2.1
RAM - 1GB + 512MB (1GB +1GB max) DDR PC2700
PSU - Octigen 300W model 10270PSOTG ('upgraded' from original Bestec 250W PSU [in 2011?])
Nvidia GeForce 6200 graphics card in AGP slot.