Sounds like the DHCP server may be set for IP address reservations. If so, even if you set the machine to Obtain an IP address automatically, it would not be able to connect. This is because the server that is running the DHCP service has an IP address reserved for the MAC address of the NIC card. I would contact the System / Network Administrator to inquire if DHCP reservations have been configured for that machine.
You can get the MAC address of the NIC card by running the following at the command prompt:
The MAC address will be listed as the "Physical Address". If reservations have been set, the DHCP server should be handing out a specific IP address to that machine.
Have you tried to ping that machine from another computer that is on the network?
Have you tried to ping the loopback address? (ping 127.0.0.1 OR ping loopback)
If you are unable to ping the suspect machine once it has Limited or No Connectivity, or ping out from the suspect machine, there may be an issue with the NIC card.