About 3months ago my computer (Dell Dimension 4600) began acting funny. 

Randomly when I would press the power button (PB) the computer would
not turn on i.e. no lights, no sounds, no action period. I would
return 10mins later to turn it on and it would work fine (video,
drives, cards, etc) without any noticeable issues or discernable
performance problems - it would be great as usual. However, whenever I
turned it off there was no guarantee that it would come back on the
next time I tried i.e. it was totally random.

Anyway the problem got progressively worse such that the intervals
between when it would turn it on successfully grew larger and larger
until I found myself where I am today i.e. the computer will not turn
on at all, ever.

I have tried reseeding/disconnecting cards, drives, memory, etc - no luck. 
I have tried new good power sources, new power cables, etc - no luck. 
There is a light on the motherboard (mobo) which makes me think that
the power supply (PS) is working and that the connector from the mobo
to the PB is defective (also tried reseeding this with no luck)?
However Dell cannot sell me this component part in order to test my
theory out. It could be the PS but then why does the light on the mobo
light up? Puzzled :-) ??

Check the circuit/wiring for the power switch. Make sure the solder
leads on the header aren't broken. Take a volt meter to the switch and
see if it still contacts. Intermittent function like that could be a
bad lead (connection.) As my brother the auto mechanic always says.
When in doubt always check the wires. It's cheap, and if you don't you
could end up spending a hundered bucks because your a/c compressor was
unplugged.

just short the connection points ie:

1) short the power button connectors. if the computer starts, then its
the  power button. if not then

2) short the power button connector pins on the motherboard, if it
starts then its the cable

3) otherwise its the motherboard. 

note: this is assuming the power supply is good per the light on the
motherboard. if not, and this is for EXPERTS ONLY you could try
shorting the main power switch cable from the power supply. if nothing
happens then its the power supply and not the motherboard. (the power
for the mobo light and the main power are on seperate circuits).

WARNING: Be very careful! Stay grounded at all times and use something
insulated to short the connections with like an electricians
screwdriver or a piece of insulated wire. otherwise you could blow
your motherboard/power supply.

finally, most tower cases use standard power switches that you can
pick up at any radio shack or junkyard.

Hi salford7,

If there is a light on the motherboard then we can say that the PS &
the motherboard are working fine.
Strip down the system to only power supply, motherboard & processor.
Then try to switch it on. If you see anything coming up on the screen
or in the front of the computer or even in the back then that means
these three parts are working fine. Now keep on adding one part at the
time to check which part is defective. Plz switch off the system
before u add any part & then try to switch the system on again.
Also on the back of all dell computers there are three small lights
(LED's). They light up either in red or in green color. Could you
please tell me the pattern they are showing? Then may be I would be
able to help you out...
Regards,
taurean

Hmmm.....all good suggestions. I am a little concerned about
"shorting" various connectors. I am not sure what that means or how it
would be accomplished. I also don't have such tools as a volt meter.
Sounds like buying a new PB and connecter (to the mobo) from the likes
of RadioShack (or similar) might be a place to start (at least ruling
out the current PB as the problem) and won't be too expensive.

My pc did the same thing, it was the power supply.  Make sure the
watts of the PS match what is required of your PC.  I am not a PC
expert, but I know that there are batteries that could be giving some
juice for those lights you still see.  I got a cheap power supply at
CompUsa for $29, but it had to match the original PS watts.