Kerberos authentication not working with Linked Servers in SQL Server 2012
Dealing with Active Directory is always so much fun.  The single most important thing here is to realize you are dealing with distributed data that can take time to propagate across your network.  

The SQL Servers in question had their name changed as part of an upgrade procedure; we replaced an existing machine (SQL01) running SQL Server 2005 with a new machine (SQL03) running SQL Server 2012.  SQL03 was the name of the new machine when I initially set it up in the domain.  SQL01 had an existing SPN associated with a single domain account we used for several SQL Servers running 2005.  Since it is a best practice to only run a single machine under any given domain account, I created a new account and configured SQL03 to run with that account name.  After taking the original SQL01 out of service, and renaming SQL03 to SQL01, there was an SPN conflict.  

I used the SetSPN.exe utility to delete the conflicting SPN (on the old domain account) - and it still didn't work.  At this point, I did nothing further and moved on to other items.  When I came back around 30 minutes later, KERBEROS authentication was working.  I simply needed to wait for the SPN change to propagate among our domain controllers.

I used SetSPN -L DOMAIN\Account and compared that output with SetSPN -Q MSSQLSvc/Machine.domain.inet:1433 to find the duplicate SPNs, and then used SetSPN -D MSSQLSvc/Machine.domain.inet:1433 DOMAIN\Account to remove the old SPNs.