CAT5/5e/6 are what are called UTP - Unshielded Twisted Pair - there is no "shielding", unless you specifiy that you want STP.
UTP cable achieves noise reduction or elimination through the use of the "twists" - yes CAT6 will give you less degradation and interference over distance, but, for the networks speeds being used here, you won't see any benefits.
Both degradation (loss of signal strength or attenuation) and interference can create situations where throughput is reduced, due to errors being introduced in to the transmissions to the point where the link fails completely, but as I mentioned before, for the network speeds being used here you won't see any benefits.
Perhaps more important than switching to CAT6 cable, would be ensuring that the network cables are properly terminated (the plugs/jacks fitted correctly) so as to avoid a condition known as "split pairs", which allows increased levels of interference.
These networks, up to gigabit ethernet, were designed to run on CAT5, which has been superseded by CAT5e, CAT6 will allow up to 10 gigabit ethernet, which you are not likely to find outside of a data center any time soon.
Hilaireg - I've never seen a switch specified as CAT5e compliant - and that includes my gigabit switches, in fact, the only time "Category n" is mentioned in the specification, is specifying the cables required, and that is Category 5
Cliff - I could not find any D-Link D008D switch listed - but a glance at the specsheet for the D-Link DGS2208, which ia an 8-port unmanaged gigabit switch also specifies CAT5 as the minimum requirements.
Assuming that the 1ft & 15ft cables you mention are "factory made" or at the least correctly terminated, you will see NO improvement in performance going from CAT5, or CAT5e to CAT6 - and - at those lengths, even if the cables were incorrectly assembled, I doubt that you would detect any improvement.