I'm thinking of getting a new graphics card. I currently have a GTX 970. However, if it means I have to change out all my other components then I'll reconsider.
The i5 4570 will bottleneck a 970 considering if you use physx in 1080p. But in normal operation going to 1440p with a 980ti or 1070 you should be fine. Really depends on the game and what you are going to offload to the cpu. That's a Open ended question.
You can have a CPU bottleneck in some games, and not necessarily much of the game. It VARIES. What's important is you'll have a great experience, and even if you got 15% lower FPS than someone with a much more expensive CPU who cares?
So let's just say YES it can, but it's basically a non-issue.
There's no definite cut-off where it's pointless usually. A GTX1080 will be the bottleneck when paired with an i5-4570 almost all of the time, for most games. Exceptions include poorly coded games like F4, some MMO's due to lots of client calculations, Starcraft 2 due to the battle calculations etc.
(SC2 will usually be GPU bottlenecked then flip to a CPU bottleneck during heavy battles)
*So TWEAK the game settings for the best experience. That's always important unless you want VSYNC at 60FPS (60Hz monitor) and everything maxed never dips below 60FPS.
Setting like Hairworks where it really should be coded for the GPU to process is offloaded to the CPU unless you can get the Game to run with a dedicated GPU to Process the physx calculations. So you would need two GPU's.