I have three Raspberry Pi Zeros (1.3), all between 3 and 4 years old. On two of them, it usually takes several tries of connecting and disconnecting power to get them to turn on. In about 1 out of 5 attempts, the ACT LED comes on immediately after plugging in the cable and the Pi boots up normally, and every other time the Pi draws about 50-70mA from the 5V supply but the LED does not come on and nothing is output over HDMI or the serial port.
I am able to replicate this various combinations of two Raspberry Pis, two SD cards, and various power sources. The issue occurs both when powering through the USB port and supplying 5V directly to 5V and ground pins on the header, so it is unlikely to be a faulty USB cable or connector.