Significantly depending on the game. I absolutely despise when people make assumptions that the FX-6300 won't bottleneck if you merely overclock, and they say so based on zero experience. I have a GTX 770, noteably less powerful than an R9 290, as well as an FX-6300 OC'ed to 4.2 GHz, and while on certain games my frame rate never drops and my GPU is constantly working at maximum potential because the game is more GPU bound than anything else (Crysis 2, Battlefield 3, Alan Wake) on other games the FX-6300 will create a very noticeable bottleneck since these games are CPU Bound (Crysis 1, Borderlands 2, Skyrim).
On the CPU bound side of things, Crysis 1 is hell. I've heard other people with an FX-6300 say otherwise but it's simply not happening for me. While monitoring GPU usage I can clearly tell that something is amiss since my GPU usage is constantly low in Crysis 1 and my frame rate jumps anywhere from 25 - 40 fps. In Borderlands 2 I'm usually at least close to 60, but there will be certain areas that just plummet me to as low as 30 fps (and in Borderlands 2, 30 fps looks awful). In Skyrim, I'm almost always at 60 fps with occasional bouts to 45. All of this due to bottlenecking. I'm also overclocked to a very stable 4.2 GHz and it's done very little to remove these instances of low frame rate. At most I've gained 4 - 8 fps through overclocking to 4.2 GHz, so you should know this for certain: Disregard what anyone says, especially those who do not speak from experience: The FX-6300's bottlenecking will not be "minimal" simply because you overclock. It will be there, and it will be painfully noticeable.
In other words, with some games you won't get any perceivable bottlenecking because either the games will be GPU bound, and hence the GPU will maximize it's potential first, or the FX-6300 will only start bottlenecking your GPU above 60 fps so you won't notice it anyway. However, in other games, you'll notice a lot of frame drops of varying magnitudes, mostly with open world games which are usually more CPU bound.