Well, here's what I would do:
Are there already Cat5e cables running from all the servers to switches / patch panels?
Are you having any physical layer connectivity issues? (i.e., frame errors, "flapping" ports, throughput issues, etc.)
First, check your connections at the ports and verify you don't have any higher layer issues. If you narrow it down to the cable itself, then replace the Cat5e with Cat6.
It's working without problems. Leave it alone.
Don't already have any cables connecting everything?
Go ahead and run the Cat6 (Cat6a). It's more of a pain to work with, but you'll get some practice with it and get better at it and you'll have the best cable you can buy installed for future proofing. It's not really that much more expensive.
In our data center, I've got a mix of Cat6 and Cat5e. The Cat6 was used for newer devices I've added since the rack was originally installed. The other stuff still has the Cat5e because it's still working just fine and I've got this thing about fixing things that aren't broken: I don't do it.
Just my 2 cents worth ...