Suggest start with seeing if the problem is the Power On switch, or the Power Supply.
To do this you bypass the Power On switch.
This has nothing to do with the Power On switch directly, nor it's wires.
Dell Support > Optiplex GX270 > Service Manual,http://support.dell.com/support/edocs/systems/opgx270/SM/EN/index.htm
Click on System Board
The connector number 18 - power connector (POWER), is a
20-pin ATX main power cable, connector on the motherboard.
Number 19 for the Small Mini-Tower model motherboard.
A general example of a 20-pin ATX main power cable, and it's respective connector on the motherboard,
Observe the photo to the Right.
Note the Green wire, and all of the Black wires.
The Green wire is the Soft Power On wire.
ALL Black wires are Ground wires.
Even if I am mistaken about the ATX main power cable, and it is a 24-pin ATX main power cable, it still has a Green - Soft Power On wire, and many Black - Ground wires.
Just like the 20-pin ATX main power cable, and it's connector on the motherboard. Scroll the above page down a little more ]
The Green wire is TEMPORARILY jumped to ANY Black wire.
The preferred jumper wire is a straightened out paperclip, bent into a U-shape.
The top of the U is wrapped a few times with black plastic electrical tape. This taped area is for your fingers, and thumb to hold onto.
Turn the U-shaped jumper wire upside down. The 'legs' are what is used.
One 'leg' goes down into a socket hole with the Green wire.
The other 'leg' goes down into a socket hole, with ANY Black wire in it.
ATX main power cable is plugged in, as shown in the photo to the Right.
Where the wires go down into this main power cable's connector, is the BACK of the connector.
The jumper wire goes down into the Back of the connector.
The leg of the jumper wire slides past the insulation, of the existing wire that is already in the socket hole.
The leg of the jumper wire MUST go down into the socket hole, far enough to touch a metal terminal, that is on the end of the wire.
Looking at the middle photo, you are looking at the tips of this style of metal connector, that is on the end of every wire, that goes in the 20-pin ATX main power cable's connector,http://www.molex.com/molex/products/datasheet.jsp?part=active/0002081202_CRIMP_TERMINALS.xml&channel=Products&Lang=en-US
The left side in the view is the part that crimps around the insulation of the wire, and the part the U-shaped jumper wire touches.
The right side of the view, shows the tip that you see in the socket holes, shown in the middle photo.
There may be a spark when the U-shaped jumper wire touches those metal terminals. (Green wire/Black wire)
You may wish to wear a glove on the hand, that you are using for the jumper wire.
HOWEVER, the Soft Power On circuit uses 5 Volts DC.
In comparison two D cell flashlight batteries store 3 Volts DC.
The glove thing is just to make you feel safer.
The contact made is no more than 2 seconds.
The Power On switch is a Momentary Contact Switch.
IF, the Power Supply comes on you have a bad Power On switch.
IF, the Power Supply does Not come on you have a bad Power Supply.
No if's, buts, or doubt about it.
Post back with your findings in a Comment. Also post what type model it is.
Small Form Factor (SFF)?
The Power Supply case shape is what differs.
You may be able to use this type of Power On switch, if yours is bad. It will be part of the plastic Power On Button assembly.
(After removing the Power On Button assembly, I use warm water, (Almost hot), or a hair dryer, to soften the plastic Power On Button assembly, then replace the Power On switch )http://www.directron.com/atxswitch.html