"Or are there games that have issues with 4GB VRAM?"
4GB is typically enough for just about ANY game. But don't just believe me I found an article that is highly relevant to your question.
This might be the perfect article snip for you!
"Results
1080p: At 1920x1080, we're seeing an average of around 1.5GB to 2GB of VRAM being used, with GTA V blowing all the way out to 3.7GB, and Shadow of Mordor getting there with 3.3GB. Still, 1080p is quite tame on the framebuffer.
1440p: Moving up to 2560x1440, we see GTA V consume 3.9GB of VRAM while Shadow of Mordor uses 3.5GB. Thief begins to crawl up there with 2.9GB of VRAM, while Battlefield 4 on the Ultra preset (minus AA) is only consuming 2.2GB of VRAM and is one of the best looking games out right now.
4K: As for what games or tests use the most VRAM, we see that it's a tie between Thief and GTA V, both using 4.3GB of VRAM, while Shadow of Mordor isn't far behind with 3.9GB being used - all at 4K. Battlefield 4 only uses 3GB of VRAM, which is quite surprising.
Final Thoughts
Surprised? Probably. 4GB of VRAM is more than enough for most video cards today, even at 4K. We haven't taken into account any anti-aliasing, as we're going to follow through with another article that looks at 1080p, 1440p and 4K with 4xAA enabled to see how much AA strains the framebuffer in these titles. But in all my years of using PCs, I barely use AA. AA is a personal preference. I'd rather have high framerates on my 120-144Hz screens, but anti-aliasing really helps at 1080p and below."
Read more: http://www.tweaktown.com/tweakipedia/89/much-vram-need-1080p-1440p-4k/index.html