The problem was that a virus had integrated itself into the TCP/IP stack and was intercepting all name resolution requests.
Even though this initially appeared to be a DNS problem, the virus was ultimately to blame.
Another thing to look at is whether the problem affects all of the users on the network or it's limited to a subset of users.
If you determine that only some users are affected, check to see whether all those users are located on a common network segment.
If you can ping the host by IP address but not by name, check your DNS server to make sure that a Host (A) record exists for the host.
Without a Host (A) record, the DNS server will be unable to resolve the host's name.
If your DNS server does use forwarders, you can try pinging the server to see whether it's online.
You might also have to call the ISP to see whether it's having any DNS issues and to make sure that the IP address you are using in your forwarder is still valid.
Even though many DNS servers use root hints for Internet name resolution, some use forwarders to link to an ISP's DNS server.
And if the ISP's DNS server goes down, Internet name resolution will cease to function as the entries in the resolver cache expire.