For almost all current games out there, 2GB of VRAM is enough for 1080p. While some games have shown more VRAM usage than that, the usage in those games can be kept under the 2GB limit by lowering anti-aliasing settings.
Also, the GPU will run out of raw horsepower at graphics settings which require significantly higher amount of VRAM than 2GB and will not be able to produce playable framerates. So considering the raw horsepower of a 680, 2GB is plenty enough for gaming at 1080p.
Hope this helps! Nice avatar btw!