Can I use CAT6 cable in a CAT5e environment? (switches / patch panels)

It should work.

You're installing a higher grade cable, but all that will happen is that your equipment will treat it as though it were CAT5e (assuming it can tell the difference of course).

The only issues might arise in the future when you have higher spec routers, switches and computers that can make use of CAT6 but half your network is still CAT5e. The network will run at the speed of the slowest component. How much of a problem that is will depend on what you are doing of course.

The difference between Cat5e and Cat6 is in insulation and quality of cable, as such if you follow the same wiring guide it will always work in your environment. Category cables can be purposed for many different things, not just networking. The defining elements are in how it is made and the insulation quality, not in the wiring patterns you use.

In fact, I actually recommend you use just Cat6 going forward. The price difference is marginal, the quality of signal is better, and if you use it with gigabit networking you will see more consistant and higher transfer rates. I run a LAN party larger-scale event and we ONLY use Cat6 as we have a 100% gigabit network. The cost difference is laughable, and because we re-use the cabling, it is totally worth it.

EDIT: Cat5 cable can actually use more than 4 twisted pairs of wires, I assume Cat6 is the same. My previous claim that the wiring count is static is inaccurate, however most wire you can buy is only 4 twisted pairs of wire.

Don't think you need to tear out your Cat5e cabling. Cat5e will support gigabit speeds which is sufficient for most homes today and for the foreseeable future. Higher speeds such as 10 GigE require Cat6.