Context Menus Not Appearing When On Top of Desktop

Weird PC question here.
I have a rig in my lab running a proprietary app and Windows 2000.
I have a number of applications running in the system tray.
In some of those, I am expected to access their menus and/or shut them down by right-clicking on their icons in the system tray.
Normally a 4-item context menu pops up, and I select from Shutdown or one of 3 other options.
I looked at the unit and today, when I right-click on the menu, nothing shows. If I right-click and then move the mouse upward, though, and the context menu appears, but ONLY the single command found in the context menu that I am mousing ever appears. If I move up, I see the next one up, but the one below goes away and I see the desktop rather than that element of the menu.

Anyone ever seen anything like that?

PS: Yes, I know Win2K is old and sucks. This entire rig is running proprietary apps which the vendor will not be upgrading to Win2K.