My Intel Q6600 bottlenecked my single 560Ti...so I would say yes, your processor will bottleneck you card (indirectly).
If we're only talking BF3, then your procecessor is at times bottlenecking your entire system. BF3 is quite CPU intensive and regardless of you GPU, will output lower frames depending on the speed and the number of cores on your processor.
To go further, BF3 Mulitplayer will really test a system (say you go in the biggest map with 64 players). CPU handles all the logistics (from player positions to physics, etc.), GPU handles the textures and effects, and HDD/SSD will shoot textures to your RAM/VRAM. All this to say that BF3 will maximize most components on your PC.
Side effects:
CPU: when entering a full server (64 people), there's a lot of movement/physics occuring at once. Furthermore, there's generally a lot more going on at once and therefore will stress your CPU more than a typical single player level.
GPU: If set to higher settings, the game needs to load->transfer->output textures, effects (code OLE from DX and the Frostbyte engine). All these will stress your GPU and playing with the settings should help you get the perfect visual versus FPS (Performance). AMD or Nvidia usually have recommendations based on the GPU model you have.
HDD/SDD: On bigger maps, although most textures a loaded on to your RAM/VRAM, it's impossible to load everything. So, occasionally, your PC will read from the source (your hard drive). If you have a slow HDD then you might want to consider an upgrade. Best way to figure out whether you're HD is slow or not is by the time which it takes to load a level before playing. If it's ridiculiously slow, then you have your answer. I've gotten a SSD drive for improving loading time and it immidiatly reduced the amount of stuttering I was previously experiencing in games like BF3, Skyrim, and other open world/huge map games. Highly recommended upgrade!
That said, I don't have an answer for you. If I had to choose one upgrade, it'd be the CPU (but that means a new motherboard, possibly ram, etc.).