A GTX 770 or 680 (same GPU) with 2GB VRAM, which was about 4 years old when BF1 launched, will still play that game at 1080p high at about 60 fps depending on map. BF1 uses about 1,9GB VRAM on high settings. Turn the settings up to ultra though and fps will drop down to ~35 average with lots of small "hiccups" and frame drops when it needs to load textures from the system ram because there's not enough VRAM.
The 4GB versions of those cards however, will hover around ~50 fps depending on map at 1080p ultra.
There is very much a VRAM bottleneck going on in this case.