I had a 350w PSU in my computer and when i added 16gb ram, and my gtx 750 the computer started randomly cutting off. after the second time i popped the side off and it was very hot inside. The PSU was definitely at its limit.
The rest of the set up was 2 HDD, and an AMD A8 APU.
I put a 750w in mine but you could probably get away with a 500w.