IP config information on the computer shows an address received from a DHCP server at 10.0.1.1 with a Class C mask, and the IP address given to the client is 10.0.1.x with a Class C mask. Default gateway is 10.0.1.1. There are no other DHCP servers set up on my network.
This is tough to troubleshoot without more information. The problem with accessing the domain controller is intermittent, right? When they can't access the domain controller, can they ping the IP address of the domain controller? If not, then it's a network connectivity problem. If they can ping the IP address, then it's a name resolution problem (probably WINS, since you're using NT4).
To me, it sounds like a name resolution problem. Make sure you have a WINS server configured, and that the DHCP server is assigning clients an address for the WINS server. Good luck.
This was first published in June 2002