It really does depend on the game and also as strangestranger says more importantly the resolution. More demanding games with more complex or heavy shader loads will benefit from more ram. Also the higher you go with the resolution the more you are asking the card to do(obviously). Think about the older GTS 320 or 640. the 640 card would fair better at higher resolutions where the extra ram would come into it but at lower res they were near enough the same card.
At the same resolution I doubt it. What card are you wanting? Might consider a couple of budget cards depending what you want. Newegg had a 9600gt for 150 recently, or an 8800 gs which should also do good, I think it was 140 but had a 30 dollar mail in rebate.
High resolution textures can take up lots of VRAM. Some games (Ghost Recon: Advanced Warfighter 2 for example) won't even let you select "high" textures unless you have 512MB of VRAM.
Also keep in mind that low resolutions such as 1152X864 will benefit greatly from anti-aliasing, which requires a lot of VRAM.
At this point I wouldn't recommend a video card with less than 512MB for gaming. Games are only going to get more demanding, and 512MB cards aren't very much more expensive than their 256MB counterparts.