CAT5e and especially cat6/cat7 is not usually hair thick, it's usually 24 AWG (and in better cables 23 or even 22). The ampacity of 24AWG is around 3A (more if you accept a bit of heat). PoE is limited to less than 1A per pair, but it makes the power through voltage (usually between 48 and 57 volts), and by using more than one pair.