On the machine experiencing the problem, in the Computer Name tab of the System Properties dialog (brought up from Control Panel | System), the "Network ID" and "Change" buttons were disabled (grayed out). A message at the bottom of the dialog indicated that the reason for this was that "networking is not installed or properly configured".
I was able to fix the problem by removing and re-installing the machine's Client for Microsoft Networks, as follows:
- From the Control Panel, open Network Connections.
- Right-click on my active Local Area Network connection and choose Properties.
- In the Local Area Connection Properties dialog, select Client for Microsoft Networks, then click Uninstall. Close the dialog and reboot when prompted.
- After rebooting, bring up the Local Area Connection Properties again.
- Re-install the Client for Microsoft Networks by clicking the Install button, selecting Client from the Select Network Component Type dialog, then selecting Client for Microsoft Networks from the Select Network Client dialog. Click OK all the way back out, then reboot the machine once more when prompted.
I don't know enough about the internals of Windows networking to understand *why* this solution worked to fix this problem in my case -- only that it did work! I share the steps I took here in hopes that they might be helpful to others as well (as a possible alternative to more extreme solutions such as re-installing Windows!).
(Please do consider carefully before following these steps; Client for Microsoft Networks is a critical Windows networking component, and removing it -- even with the intention of subsequently re-installing it -- could possibly have other adverse affects on a system. For example, if the re-install of the Client for Microsoft Networks were to fail for whatever reason, the machine could be left in a state where the machine's network connectivity is even further adversely affected.)