Depends on the speed of the CPU, it may a bit, but usually bottlenecks are over exaggerated and they are not as big of a deal as people say. It depends on the game, and it depends on your CPU speed. My CPU (a phenom IIx6) is technically a bottleneck but I only notice it in games like farcry 2 and crysis and it is not a big deal, it is the drop to about 50 FPS instead of getting above 60 all the time (i usually hit 90) it just hits sometimes. It should be fine, and you will notice a increase, just not a HUGE increase.
I don't mean to start the flame war but yes, the phenom would be a bottleneck of the computer. (Sorry) I would recommend at least an i5 to keep up with those puppies, also the sapphire brand for that gpu is really hit or miss 50% of the cards are lemons try a different brand ie: Asus or Gigabyte. I mean i guess it really depends on what games you are playing, if you play something like Planetside 2 it's a cpu hog and the cpu will be a bottleneck but if it's for something like WoW then no it will not bottleneck.
I know how games work (and how to make them for that matter) so I may be able to answer this question.
Before the graphics card can render anything the game needs to update (and re-sync with how long it's been since the last update). The update can be bottlenecked by the CPU because one of the things is that it's not easy to make the update multithreaded because things just need to go in order, not threads going all over the place doing there own thing. So it's good to have powerful threads (the Intel ones) in order to update the frame. The update depends on the game your playing, which things like battlefield 3 will need a good CPU to update every little thing in the game from a plant moving to decoding the packets (if your on multiplayer) which can also slow it down. The games should have nice and fast cores to be able to run as fast as it can although slow CPUs lag can be hidden if the graphics card cannot keep up with it. Keep in mind in the most intensive games normally %100 CPU usage is never seen as other things such as RAM keep it from going fast. Overall you need everything to be fast and stable, this means lots of fast RAM, good MOBO, good CPU and good graphics card. Any of these can limit your game's performance but with your CPU if you overclock no problem!