Ok, I'm surmising that it has something to do with a PC hosting VM's. There is one more PC that we can't ping and have trouble accessing, and it too hosts VM's.
In the Server Manager consoled, I notice that both these troubled machines have multiple entries in the DNS Server->Forward Lookup Zones: One entry for the physical NIC, and two more for the VM virtual adapters.
EDIT: I don't think this is the issue. None of the PC's that host VM's here in my home lab are exhibiting this problem.
