Is this Mac OS X Server 10.6 Snow Leopard Server, or Mac OS X client 10.6 Snow Leopard Client?
In my experience, the dig command works fine.
The command response indicates there are no DNS servers reachable.
In general, the dig command asks your designated DNS server for information, and errors imply that the (remote) DNS servers are misconfigured or unresponsive, that the network is misconfigured or that there is a routing error or that a firewall is blocking access, that the dig command syntax was wrong, or that the (local) DNS server configuration is incorrect.
You'll want to indicate more information about your LAN configuration, whether you're working with static or dynamic IP (servers want static IP), whether you're running your own DNS server (servers tend to have that running), and what sort of IP routing is set up here (firewalls? direct connect to the modem?) and whether the DNS server and IP gateway is registered with your network controller.
If you're running Mac OS X Server, [here's some reading material on setting up DNS services on your server|http://labs.hoffmanlabs.com/node/1436].
