Look at the graphs above, the 560TI 448 edition performs better than a 6970 2GB, not just average FPS but also min FPS, yet it doesn't have 2GB of VRAM and therefore makes the whole debate about 1GB VS 2GB for 1080P is worthless as there are more important things to take into consideration.
Not too sure if you are talking about the normal 560TI with regards to it being better than the 6950 2GB, if you are talking about the normal 560TI, then that most certainly isn't the case, the 6950 1GB/2GB is better than a normal 560TI, the 560TI 448 edition is better than a 6950 2GB/6970 2GB and pretty much on par with the 570, but can beat a 570 when overclocked.
VRAM on a single GPU set up does not matter one bit when it comes to a res. of 1920x1200 or below (maybe on the less powerful cards like the 560TI 1GB etc. when everything is FULLY maxed out, but a card like the 560TI is not meant to run the game at those settings and at a res of 1080/1200, you need either a 570 or 580 or a dual GPU setup at 1080/1200).
Emmm, what?
A 6970 2GB on par with the 470?
http://www.anandtech.com/bench/Prod...0.213.216.219.221.238.228.229.231.232.234.235
Again a single 470 is not meant to run BF 3 maxed out (maybe if it had more VRAM, it would have helped with min FPS when you have 4xAA set). The reason it is smoother than your 470 is not because of VRAM but due to the 6970 being a more powerful card in nearly every area and the 6970 is capable of running BF 3 pretty much maxed out at 1080P.
Also the reason it may have been terrible on your quad SLI setup is because drivers/game isn't optimized well enough, just like many other games out there that are crap on systems that have more than 2GPUs.
I have seen plenty of people with 6870 crossfire and 470 SLI that run BF 3 perfectly fine at 1080 all on ultra with AA set to 2.
EDIT:
And one last thing, benchmarks mean nothing when it comes to real world performance.