If your problem is with a OpenType font — that issue has been covered before (and in other forums as well). Here's the gist of it: older versions of Quark do not support OpenType-faces. Only the first 256 or so glyphs are recognized. Solution: update to QXP 7 or go the Adobe route (InDesign rocks!).
Possible solution: do not use OTF but only PS (with Expert-sets).

The problem is occurring with both OT and PS fonts. What happens is I get the Light and the Medium weights to show up, and not the Bold and Regular weights. It doesn't make sense to me.

> I’m having an issue where only part of a set of fonts becomes active

Are these fonts something that you made or were they licensed?

I guess you're problem is related to the way font families were merged and linked into one suitcase in the OS 9 days.
Try this to hopefully solve the problem: find your original font files (with separate suitcase files for each variant) and use those.
Or else use Font/DAMover (a Classic app!) to split up the faulty font suitcase(s) into separate suitcases for each subfamily.

Or it can be FontExplorer's problem. I stopped using it completely. Another pile of junk on the OSX font manager heap. IMO.