If I replace a cat5 cable with a cat6 would I see an increased data transfer rate between two systems?

no.
you cant .
CATEGORY5 and CATEGORY6 cables are almost same just used to Indoor and outdoor..
it depends on what cable you are using as well as NIC.
liek ETHERNET then u can get10 MB/s transfer rate, FastEthernet then you can get100 MB/s and GigabyteEthernet then you can get1000 MB/s so u have to choose correct cable and NIC for faster transfer rate So if you have cable ETHERNET then even you have GigabyteEthernet NIC then max transfer rate is10 mbps,, like wise you have to choose correct cable for correct NIC Thanks

 The general difference between category5e and category6 is in the transmission performance, and extension of the available bandwidth from100 MHz for category5e to200 MHz for category6. This includes better insertion loss, near end crosstalk (NEXT), return loss, and equal level far end crosstalk (ELFEXT). These improvements provide a higher signal-to-noise ratio, allowing higher reliability for current applications and higher data rates for future applications.  If the components are category6 compliant, then you will be assured of category6 performance.