I suspect the problem is some how related to a Browse Master issue. Check the event logs on the PCs and see if you see any alerts regarding a master browser election. It would appear that your Old Dell (whatever operating system it may be running) has seized the role of master browser and is the only computer on the network that is maintaining a server list often a Windows 7 machine cannot assume the role of browse master if a Windows XP machine is already up and running on the network when the Win7 machine is booted.
Maybe do some googling about master browser elections and see what you come up with.
You can, from a command prompt type
net view (to see all the computers on your network and then
also from a command prompt type nbtstat -a computername (for each computer name on the network)
look at the results for a line that has _MSBROWSE_
that should indicate which machine is the master browser for your network (workgroup)
multiple machines attempting to perform the role of master browser can result in a browser war, which Win7 machines can't seem to win when competing with XP machines, (I don't know about Vista machines)
The following registry key
HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\services\Browser\Parameters (MaintainServerList) is where this is controlled. A value of Auto supports and allows browser elections.