Does this happen in all games?
Does the FPS drop and stay down or does it fluctuate between high and low FPS?
First thing that comes to mind is that something is overheating and so its throttling performance so it doesn't kill itself, which explains why performance is fine until 10mins in. You might want to monitor temperatures while having the rig on an artificial load, download Furmark (GPU) and Prime95 (CPU) and see what temps your getting.
Here are my results, http://prntscr.com/2gcy16
When I play the temperature only goes up to around 40 at max.