Somewhat hard to explain but basically the scenario is like so:

I have an Active Directory Computer named Server1.testing.local
If I type \\server1 into the file explorer window the shares show up fine
Lets say I want to have another server name for that server like sweetserver.testing.local
I create the proper CNAME for sweetserver.testing.local into the DNS pointing to server1.testing.local
I then go to \\sweetserver and this comes up even if you enter the correct username and password:

What needs to be done is add an alternate name to the server with the following command:

netdom computername server1.testing.local /add:sweetserver

Once I did the above command in and elevated command prompt \\sweetserver works!

If you get an error that the command failed then open services.msc

Stop any SQL services running and try the command again.  Once successful start the SQL services back up.

No reboot was necessary.