Is this your home network or do you have a domain?
* If its a domain, go to your DNS server - do you have a static record for the IP/computer entry?
* If its a standalone computer and no domain, do you have a static record in your hosts file?
You say that even tho your computer is offline, you still get a ping with that IP address AND the DNS name? If there was a duplicate device, I can see the IP giving back a ping but not the DNS name
One other thing, have you tried doing a simple ipconfig /flushdns and then ipconfig /registerdns to flush the DNS cache?
Lastly, if you have a bunch of Windows machines on your network, what you could do if you're still getting a ping from this IP/box - simply try RDP'ing to the IP and or computer name. If its some live box with a duplicate, once you've logged in, you
can see in the Properties what the computer name is to track it down from there.