No. *good* Cat5 or better Ethernet cable is between 22 and 24 AWG to meet specification. Short patch leads may be as thin as 26 AWG. If its thinner it wont meet spec.
With 8 wires in an Ethernet cable, one can afford to parallel two for each conductor, so the possibility of 26 AWG isn't a problem. Assuming you are retaining the 8P8C modular connectors (colloquially 'RJ45') so you can use any off-the-shelf straight through UTP Ethernet cable, then, for idiot-proofing, some time spent studying Ethernet PoE standards is advisable to choose a pinout that will let your application survive being connected to an active PoE port, and that wont blow out any NIC the idiot attempts it connect it to.