Win98SE based computer requires 3-5 restarts to get a successful boot

I have a WIN98SE based 500MHz PentiumIII PC with 256MB RAM.  I find that the computer requires me to restart it 3 to 5 times each morning before it finally boots up.  I have addressed this to the hardware section because during the unsuccessful boots it doesn't seem to have any response (i.e. no beep noises, no hard drive or floppy drive access, no video) aside from the power supply fan and cpu cooling fans turning on (the cpu cooling fan operates from a power plug on the motherboard).  The strange thing is that after restarting 3-5 times, it will successfully boot and runs perfectly all day.  I have run virus scans and spyware scans and they both show as being clean.  I have removed and reseated all of the power and data cables to the motherboard, hard drive, floppy drive, and CD Drive.  I have removed removed and reseated all of the cards (modem, video, NIC).  I removed and reseated the memory modules.   I was thinking that maybe it was thermally connected, that is, that it had to go through the process of heating up a bit before working, but if I try a restart immediately after shutting down the system which has been running all day, it still requires 3-5 restarts.  Does anyone have suggestions on what might be the cause of this?
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This sounds like it could be the power supply failing or not being able to provide enough startup current.  The biggest load on a system is during a cold boot, and it looks like you should replace the component.

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What happens when it doesn't boot?

If you get no nothing at all perhaps your switch is going bad...
bad capacitors

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I would look at a combo of Callandor and compfixer101 suggestions. Either or both could cause this problem. I have also seen drives cause these problems but it is very rare. If the PSU and/or Caps look good, try unplugging everything but the HDD and try to cold boot it a couple of times to see if it does any better.
I think all 3 are good suggestions to check. It doesnt sound like a sw issue. rather hw.
check the power you need here :      

the boot-up process may need too much power.

and check also the bios battery, it should read 3 V
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