The presenting problem is that Windows NT appears to automatically decide which adapter to use for all network traffic. This seems to be governed by some sort of load-maximizing algorithm. To make sure that this does not happen to you, it is important to set up your ethernet adapters with care:

For any adapter connected to the internet:
 - specify IP address
 - specify subnet mask
 - specify the default gateway

For all other adapters that do not go out to the internet, but are instead
used as "private" networks,
 - specify IP address
 - specify subnet mask
 - LEAVE DEFAULT GATEWAY FIELD BLANK

Also, make sure that the "IP Forwarding" option is turned off.