I run a GTX 960 2GB on a 1080p TV screen and can run on medium to high details for most existing titles getting great frame rates for that card 50-60fps average depending on settings (of course there are exceptions). I don't get texture errors. I also don't expect it to perform as well as my gaming system running with a GTX 980 in it using a 2560x1440 G-sync monitor. I am satisfied with the performance of the GTX 960 at 1080p though and it's paired with an i3-2100. It's still better than any current-gen console.
Most of the latest titles recommend more VRAM though even if you don't have more to offer, the game is still not going to cause the driver to throw any texture errors at you. The game will run, but just at a lower framerate or you just run with less AA and fewer details.
Also, if you're worried about optimal settings, even though I think the software is clunky, Geforce Experience can give you a good starting point for video settings with whatever card you're using so there's less trial and error time with regard to the settings in any particular title.