What many people fail to realize is that because a LED lights up doesn't mean that the PS is actually working.
It just means that the circuit that the LED is connected to is getting some power and has no reference to the Current or actual Voltage actually.
LED's have a Working range between 1.5 to 12 V DC so any Voltage within that range will light the LED but doesn't indicate that the PS is actually producing the necessary Voltage or even any real current.
Most Power Supplies have 3 distinct voltages 12 V DC, 5 V DC and 3 V DC the 12 V DC is used to power things like HDD, Video Cards and so on, The 5 V DC is used to power the lower voltage items like 2.5 Inch HDD and USB Sockets. The 3 V DC is used to power the Logic Circuits so if one of these voltages is missing the Computer isn't going to work but it's quite possible that the LED's in the computer will light up.
The Green LED that you are referring to is most likely the one on the Network Connection telling you that the Network is connected so that can mean anything from a Ethernet Cable to a Switch or Modem is connecting to the Switch or Modem and is getting somewhere between 1.5 to 12 V DC. Also because LED's draw very little power if the current has gone way off and the PS is only producing very little in the way of Amperage the LED's are still going to light up and look as if they are working. But things like Fans and HDD's are unlikely to work in a situation like that.
I would suggest swapping out the PS with a [b]Known Good[/b] one and see if the thing work now but in the mean time you can try unplugging the Power Lead and pressing the On Button for 30 Seconds. This will discharge all of the Capacitors in the system and then plug the Power lead back in and try turning on now.
If the system starts the Electrolytic Capacitors in the PS have or are failing and the PS needs replacing.
Col