How to increase the Maximum number of connections to a remote computer?

I have a lab of new Vista (Business) machines.  I'm sharing a printer that's connected locally on one of the (Vista) PCs.  The machines are all in the same workgroup.  In order to avoid any issues of not being able to "see" the remote printer, I have a script that runs at login on each machine, creating a mapped drive to a share on the machine that has the printer.
On only a few machines, I'm getting the following error when I try to map the drive:
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.
What is causing this error and how can I increase that limit?
Thanks
(The entire error message is copied below)

>>>>> Start of Error message copied from the CMD window<<<<<
Microsoft Windows [Version 6.0.6000]
Copyright (c) 2006 Microsoft Corporation.  All rights reserved.

D:\>net use u: \\d003\downloads /user:teacher 2008
System error 71 has occurred.

No more connections can be made to this remote computer at this time because the
re are already as many connections as the computer can accept.
>>>>> End of Error message copied from the CMD window<<<<<