Too many networks connections

We've got a Windows NT-based network spread over 5 cities, with about 40 servers altogether, and a lot of shares (several thousand).  Servers are running NT 4.0 SP5.

Some of the Win95 PCs seem to go into some sort of discovery process just after they log on.
Most of the time, its not a problem, but some appear to hang on to the shares they find, and eventually end up with 256 Network connections.  Then, when someone tries to map a new drive, or printer on their PC, they get the message "Too many networks connections"

I found this "discovery process" by going into a DOS prompt, and typing:
NET USE
The PCs that exhibit this problem have a few "normal" connections:
Status        Local name    Remote name
OK            M:            \\ELRB0055\AutoCAD

and a whole bunch of unnamed connections:
Status        Local name    Remote name
OK                          \\ELRB0079\X_Drive

It looks to me like the "unnamed connections" is a listing of every single share on our network.  It goes through all the shares on one server, then the next server, sometimes hits any user's PCs that have their hard drives shared, and eventually just quits after 256 total connections.

On some machines, it does this immediately after logging on, causes a problem for about a half-hour, and then goes away (No more unnamed connections).  One some other machines, I get a blue screen of death when it hits 256.
I can watch shares get added slowly to the NET USE list, and, if I catch it & remove them fast enough I'm OK.  If I wait more than 5 or ten minutes, I get the blue screen.

We're running Norton Anti-Virus 5, and McAfee Remote Desktop 32 client on these machines, if that matters.  I have, however removed those programs from the StartUp, and had the same problem.

Anybody got any ideas?