So a domain admin gave the account permissions to dynamically modify spn's with read and write. This fixed an issue we were having where people could not log in to SSMS from their machines. We did for another account as well, the one that we are having the linked server issue, but this did not resolve the problem. We read somewhere that allowing the computer to delegate the account would fix the issue, so we changed this setting for the machine on the computer object in AD. We still get a failed test to linked servers from our SSMS. If we log in to the machine over RDP and then open up SSMS and test the linked server connection, it works. It get's confusing though because I think when I test it, it's actually using my domain account and not the service account.