armyman your PSU isn't enough. You shouldn't run a PSU at more than 80% of it's rating. rx480's can pull up to about 225 watts each, which is 1,350 watts. You have up to another 150-200 watts for the system also, excluding the vid cards. So you are over 1500 watts on a 1200 watt PSU. Of course it will not be stable. A 1600 watt PSU will run 6 480's and the system....just. It is still pushing the limit a bit much when you factor in the 80% rule....but it will run stable.
480 draws ~165w stock (this is by card, not at the wall). If you don't change power limit, the cards would draw ~ 1000w, and other system parts maybe 50-100, so PSU shouldn't be overloaded, but is higher than you would want for 24x7 operation, and may be slightly less stable near full load. You can undervolt to save ~20-30w per card.