Notes re RAMPS:
Are you sure it's the PSU shutting down and not the PTC polyfuses on the RAMPS board?
Make sure they aren't touching anything, and make sure the fan you point at your stepper drivers also keeps them cool.
They work by heating up, and when they reach a specific internal temp, they trip. a momentary loss of power (such as turning off and on a PSU) will usually reset them.
I've seen cases where if the larger 11A one is touching the 5A one, the 5A one trips when it's nowhere near 5A usage (the 11A one runs much hotter, causing the 5A one to trip).
Personal opinion: Pull out the polyfuses and replace them with bits of wire. Put proper fuses on the inputs to your board so that you have some over current protection.
Note re ATX PSU's:
An ATX PSU doesn't always allow you to draw all the current they say it can off a specific rail. Usually it's broken up into either pairs or threes so that you can only pull some of the current off a particular wire, but in total you can pull it out of all of them. Check the current rating per 12V rail, and not the 'total' rating of the PSU.