2GB of RAM is really ideal, won't have too many issues, when doing 1080p.
With that said, with a little work adjusting graphics settings it shouldn't have too bad of a time running 1440p, the reason being that disabling features like AA drastically reduce RAM usage and demands on the rest of the card, so 1440p would be workable.
When you get up to 4k, 4GB is really the minimum advisable for gaming, because the demands on the RAM are so intesne. With a card like the R9 270x, with 4GB of RAM you can likely attain 30FPS at 4k using no AA and dropping settings to about medium. So the performance won't be so great, and at that point it will likely look worse than 1080p because of the reduced graphics settings.
With all that said, if you...