The ends of a network link will attempt to work at the speed they are configured for (if configured/configurable), or autonegotiate to the highest common speed. If the electrical performance of the link between the parties is not good enough, you get errors. Very rarely do I see switches drop down to a lower speed, and when they do, it is usuallly 10Mbps.
The switches only see the electrical performance on the link, not what is printed on the components:
I have a bit of experience with running networks on old cabling. Work has ~40 switches, and the edge cabling and termination that goes with that, all cat5 from the 10Mbps era. Not cat5e, plain old cat5. The 1000BASE-T spec allows gigabit on cat5, and it does work, but with no margin for maximum length runs. cat5e has (from memory) stricter crosstalk specifications, with leads to some margin.
As Xineas notes, as far as the 8P8C connector goes, the 'cat6' one accomodates thicker wires, which in turn allow higher PoE power limits. As far as actually crimping it onto a cable is concerned, I find that a tightly-twisted cat5e connector on cat5e cable will run 10Gbps, whereas a cat6 one with the twist of the cat6 cable unwound too far will give me errors. So take care and make pretty terminations, they matter.
If you find yourself in a situation where you are running 1Gbps on cat5 (not 5e), or 10Gbps on cat5e (which is not officially an option, as the cat5e requirements are not strict enough), I very strongly recommend some level of management on the switches. Without it finding that defective link becomes impossible.
