It depends on what resolution your display is and what 2.6Ghz dual core you have (A64x2, PII x2, Core2, P4D, etc?).
If you are running at a lower resolution with a weaker CPU, you will have a higher probability of being bottlenecked.
Try and overclock your CPU.
This will give you the maximum performance from any current or future GPU.
Regardless, the 4870 is a much more powerful card than a 9600gt.
Even if you are somewhat bottlenecked, you should be able to maintain a higher minimum frame rate and you will be able to run your games with more eye candy with no performance impact.