First, here is my current setup.
Gigabyte B150M-D3H Ultra Durable Motherboard
Corsair Vengeance LPX 16GB Ram
Corsair RM 650x Gold Rated PSU
i5 6600 3.3ghz Processor
XFX RX 480 8gb
Dual Monitors - Samsung SyncMaster 216BW / Compaq S1922a - 1 connected through an hdmi to vga, and 1 through a displayport to vga adapter, since neither monitor has hdmi nor displayport connectors. (It was a budget build. I was keeping myself under $800. No money for new monitors.)
I built the computer fresh right when the RX 480 came out, and everything was great for about a month, then I got some random reboot. I wasn't sure what was going on, but I simply turned down the settings on my game slightly and it completely went away... for about a month when it happened again. I tried lowering settings, it didn't happen for a few weeks, then it happened again. Over the last month reboots started becoming frequent no matter what I would do. I didn't have to be playing games or anything. I had a number of times where I'd simply turn the computer on, and wait a few minutes and it would just reboot. Nothing open, unnecessary programs all taken off auto start. It has been extremely frustrating as I was basically unable to do anything on the computer without it rebooting. The only reason I can get this all typed up without it rebooting is because I finally gave up and pulled the GPU out, and am currently running off the motherboard's obviously inferior graphics. After taking out the GPU I've had no problems. No reboots. Nothing.
This leads me to believe that there is only two possibilities. Either the GPU is bad, or the PSU can't handle the load of the RX 480. I currently don't have, nor do I have the money or friends with whom I can get and test out a different power supply. (You may think, 'oh they're cheap, it's not that hard to get one. blah blah blah.' Trust me. If I say I legitimately can't get a hold of another PSU, it's because I can't despite praying to the wallet gods otherwise.)
I'm just trying to get a little insight/input that could lead me to a solution that isn't going to cost me anything to remedy. I'd love to be able to use my GPU, but the constant resetting got unbearably bad.