Depends on what you consider good. In terms of Video Settings at least. The 8800GT is a 7 Year old card and Dual cores are starting to become obsolete.
And no. You might get 30FPS on Low Settings.
BF4 System Requirements-
Minimum
OS: 32-bit Vista SP2
CPU(AMD): 2,8 GHz Athlon X2
CPU(Intel): 2,4 GHz Core 2 Duo
RAM: 4 GB
HDD: 30 GB
GPU(AMD): Radeon 3870
GPU(Nvidia): GeForce 8800 GT
VRAM: 512 MB
Recommended
CPU(AMD): 6-core
CPU(Intel): 4-core
RAM: 8 GB
HDD: 30 GB
GPU(AMD): Radeon 7870
GPU(Nvidia): GeForce GTX 660
VRAM: 3 GB
You meet the Minimum, But your going to have to lower your Resolution and Video settings to make it playable.