If you can ping the server by name, then DNS is resolving the name, but SQL Server might be configured to allow connections only on a specific port. If so, you'll need to include it when connecting via SSMS. If you don't have this, the DBA should be able to provide you with the information.
I totally agree with this except for the fact that the OP said he was able to connect via IP fine and did not mention anything about specifying port numbers when he connected with IP. If he was able to connect by IP and not specify a port then name should be fine as well, unless the name cannot be resolved or the server name changed and he did not update SQL appropriately.
Joie Andrew"Since 1982"