Current over CAT6 ethernet cable
At 5V you'll probably run into voltage drop issues before you run into current limitations (if the length is more than a few meters).

Some CAT6 cable is rated as low as 60°C, and some is AWG 24, so if your ambient could be as high as 50°C. the current limitation might be as low as 2-3A. See, for example, this and this. 

Edit: If the length could be as long as 10m, and assuming AWG24 size-- resistance is nominally 84 ohms/km so 0.84\$\Omega\$/10m, so three in parallel, round trip, would be 0.56 ohm at 20°C. If 5% voltage drop (250mV) was acceptable, that would be a current of 440mA maximum, so maybe 350-400mA maximum allowing for temperature.