No More Connections Can Be Made To This Remote Computer

I've got a networking problem that's left me banging my head against the wall. First though, this is a workgroup with the following machines:

Computer 1: Windows 7 32-bit used as a peer-to-peer file server, app and DB server, designated network browser, inbound VPN server (MS VPN)
Computer 2: Windows 7 64-bit used as a workstation
Computer 3: Windows 7 32-bit used as a workstation
Computer 4: Windows XP SP3 used as a workstation
Printer/AIO: Brother Laser, network-attached

Computers keep appearing and disappearing off the network except for the computer #1. In other words, on all the other machines, when you go into Windows Explorer and click on "Network", sometimes only one machine shows up; at other times, all machines show up. Mapped drives work without fail - it is just the network browsing that is unstable.

When only one computer is showing up in the list, if you issue the "net view" command, you get the message, "No more connections can be made to this remote computer at this time because there are already as many connections as the computer can accept." I'm not sure what is eating up all the available "connections". I thought that Windows 7 allows 20 network connections, i.e., 20 computers on a LAN.

I found a machine that was joined to a non-existent homegroup, and turned that all off, but it had no effect. I've temporarily turned off any firewall rules (F-Secure) that I thought may be interfering with this, but no effect. I've also looked at the lanmanserver registry keys and I'm not seeing anything other than stock settings.