Pizzapaz001

First of all I want to say that I do not have any experience and I'm just searching on other topics to look for solutions.
 
- In this same Topic it was fixed by reinstalling windows but that I would look to some other options first because a full reinstall is the worst thing to do
 
- Another fix that helped someone with a rx 480 (only use if u had an old nvidia GPU):
Remover all your drivers (Nvidia from the old GPU and AMD from the new one) with DDU:
http://www.guru3d.com/files-details/display-driver-unin...
install AMD Drivers only.

PSU or some sort of overheating most likely since you mention it is only under heavy load. When this happens what are overall system temps? Maybe VRM's overheat?
Is your build dust free?
 

PSU most likely tough it could be RAM or the motherboard itself!

yea psu in one of the important component please don't skip out on it. 

A wonky PSU could absolutely cause issues. It's one of the things people skip on and it can bite you in the ass.

Avoid the 1060 Garbage Edition. Get a 4GB card MINIMUM

3GB 1060 has 10% less shaders and the drivers control GPU when you hit the 3GB limit now so you get performance akin to a 470 or lower in some cases. Nvidia added that to avoid stuttering but the result is MUCH lower performance

Depending on the game and clock speeds (look at @don_svetlio post)
 
And no, 4GB really is the minimum for 1080p gaming now, I've hit 3.6GB 1080p on R6S and 3.4GB on GTA 5 so...it is valid  (and they are powerful enough cards to push games at 1080p ultra so no excuses for turning settings down).
 
Also OP, look around for 4GB RX480s, they aren't that much more than the 3GB 1060/RX470s...
Germany:Italy:Britain (just as an example as once converted to Euros, it's only 230)

I am getting the rx470, I'd rather have 1gb more vram than about a 10% fps increase. 

I am getting the rx470, I'd rather have 1gb more vram than about a 10% fps increase. 

Very doubtful. The RX 470 isn't super duper powerful anyway. The CPU is the biggest offender usually only in sim games. So you should be good to go!

no not on 60fps

The i7 920 is a good enough CPU and it will be fine running a RX470.

Likely yes. Your i7 is slower than FX6 CPUs and the 470 performs like a 970

No bottleneck then. And don't run FO4 over 60hz since the physics engine is highly dependent upon the frame-rate.