Will AMD Athlon 64 X2 5600+ 2.9 Ghz bottleneck SAPPHIRE R7 260X 2GB GDDR5 OC 2x DVI Video Card??

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Unfortunately it will.

Furthermore, the PCIe x16 slot on an AM2 slotted motherboard is probably a PCIe x16 1.0 slot rather than a PCIe x16 2.0 slot. A 1.0 slot will further bottleneck the card.

Anything less than 6GB of RAM will represent a further bottleneck.

If you're considering a modern GPU then it's best to start a new build with a PCIe x16 3.0 motherboard to future-proof yourself against any upgrades you may consider over the coming years. PCIe x16 3.0 will be the standard for the coming years and as-yet has not been fully utilised by any consumer-level GPUs.

EDIT: While the below post makes a valid point about system-specs, I'm making very educated guesses as to your hardware based on the types...
Unfortunately it will.

Furthermore, the PCIe x16 slot on an AM2 slotted motherboard is probably a PCIe x16 1.0 slot rather than a PCIe x16 2.0 slot. A 1.0 slot will further bottleneck the card.

Anything less than 6GB of RAM will represent a further bottleneck.

If you're considering a modern GPU then it's best to start a new build with a PCIe x16 3.0 motherboard to future-proof yourself against any upgrades you may consider over the coming years. PCIe x16 3.0 will be the standard for the coming years and as-yet has not been fully utilised by any consumer-level GPUs.

EDIT: While the below post makes a valid point about system-specs, I'm making very educated guesses as to your hardware based on the types of motherboards that will support your stated CPU.
 
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