OK, so measuring the height of one's PC is a very poor way to gauge its GPU performance. And what are we applying the ruler to anyway? The tower? The monitor? The length of your video? It's so much easier to just fire up the game in question -- Star Wars Battlefront -- and check out the frame rate. That's what Gamers Nexus has done anyway.
And it worked out famously. The site's Steve Burke decided to put a bunch of video cards through their paces, with Battlefront providing the fire for the trial. Along with testing the most common resolution: 1920 x 1080, he also cranked things up all the way to 11.
The high end features the likes of the GeForce 980 Ti and Radeon 390X, with the low end staffed with a GeForce 750 Ti and GTX 950, with AMD's offerings being the R9 270X and R9 285.
So, what happened? There a graphs a-plenty in the article, but the long and short of it is that unless you're running a dual-980 Ti setup, 4K is asking for trouble. While the configuration managed 60fps on average, it only did so "barely".
Going down to more sane resolutions, 1440p at Ultra settings is playable starting with the GTX 970 and better. That's not to say cards like the 960 or 285 are out of the running -- you'll just have to drop the quality settings a few notches to get a smooth frame rate at 1440p.
Finally, 1080p provides a somewhat better experience for the slower cards, but you're still going to want to go easy on the shadows if you're cruising around on a 750 Ti, 950 or 270X. Interestingly, while the 4GB cards provide high frame rates, the extra memory doesn't deliver that massive benefit over those with half the amount, so don't worry too much there.
For a deeper analysis, along with a look at the game's available graphics settings, head on over to Gamers Nexus.