Wrong IP address when pinged from another machine?

This is a DNS issue. What your IT department said was true. The name of your old machine has been cached by the other machines on the network. In other words, previously the other machines went to your DNS server and asked what the IP address was of the machine matching that name. The server responded with what was the correct IP at the time, which the PC stored for future reference. This local cached copy is usually discarded after a certain period of time. You can force the purge of the cached information by doing ipconfig /flushdns on the machine you are pinging from using cmd.

If that doesn't work, then the DNS record probably hasn't been updated on the dns server. You can do ipconfig /registerdns from your new machine to try and force the update. Wait 15 minutes, go to the old machines that have the incorrect IP logged, flush the dns records again and try to ping again.

If this doesn't work, blow your IT teams mind by asking them this: 'Hi guys! Please can you check the A record for (machinename) on your DNS server? I have a suspicion it's (oldIP) rather than (newIP). If it's not, I don't think dynamic DNS updates are working properly from your DHCP server. Can you check it out and get back to me?'

UPDATE

@Kenneth In response to your update, it's entirely possible there are two A records for the same computer name in DNS. Whilst the record for your old machine may still be pointing to your old machines name, the DHCP IP address for the old machine could have been assigned to someone else. In which case, the name will resolve to the wrong PC. What you can do, is type nslookup in cmd then type the IP of the machine that is responding when it shouldn't be. This should check the reverse DNS entries instead of the A records, and should give you the real name of the machine.

In either case, you can ask your IT team to hack up DNS manually to resolve the issue. It's a really, really short and easy job. They shouldn't take any longer than 5 minutes to resolve it.