No. DHCP assigns unique addresses for each port. The addresses are actually tracked via the MAC address, which is theoretically distinct for every network adapter ever made.
If you plug in your computer to the network while also maintaining a WiFi connection, the computer will get 2 IP addresses. Furthermore, modern operating systems will assign a metric to the two connections and assign traffic accordingly. Since the Ethernet jack is usually faster, the WiFi connection will sit idle for as long as you're plugged in. (This doesn't always work, but that's how it's supposed to work.)
Usually, IP conficicts happen for one of a few reasons:
1. You manually assigned the same address to 2 computers
2. You manually assigned an address to one computer, and DHCP gave another computer the same address.
3. A computer got an address via DHCP, and the DHCP server got completely reset. Then the DHCP later handed out the same address to a different computer.