Thanks to @opc.three for all the input. If this is the case, it seems rather ridiculous that all the ducks must be in a perfect row for this to work. At best it is a pain, at worst it could be an unworkable situation. However, this still doesn't seem to add up. I have a server registered as SPN ProdA.my.dom, but only added as linked server ProdA and it seems to work fine. I have registered ProdB as ProdB, ProdB.my.dom, and ProdB:1433 and my linked server using the current user's credentials do not work on that one.