Will Intel Pentium G4400 bottleneck Radeon RX 570?

Solution
Yes, without any doubt.

However, you might be able to improve things somewhat by making sure you have the latest motherboard bios version installed, the latest chipset drivers installed, and that you have the most current drivers for your audio, network adapter and storage controllers as available on your motherboard product support page.

Also, doing a clean install of the GPU card drivers if you have not already done so would probably be an equally good idea.

http://www.tomshardware.com/faq/id-2767677/perform-clean-install-video-card-drivers.html


Honestly though, you're already probably getting pretty good FPS considering the CPU you're using.
Yes, without any doubt.

However, you might be able to improve things somewhat by making sure you have the latest motherboard bios version installed, the latest chipset drivers installed, and that you have the most current drivers for your audio, network adapter and storage controllers as available on your motherboard product support page.

Also, doing a clean install of the GPU card drivers if you have not already done so would probably be an equally good idea.

http://www.tomshardware.com/faq/id-2767677/perform-clean-install-video-card-drivers.html


Honestly though, you're already probably getting pretty good FPS considering the CPU you're using.
 
Solution
There are too many variables to know that. Anything from the resolution of your monitor, to the game quality settings, to how clean your windows installation is, to how much memory is installed, and on and on, will affect that. Any number I'd offer you would just be a guess.

Why do you need more than 120FPS? Are you running a 144hz monitor?