I have a guitar with two pickups and a three way switch. Which the switch is set to either the neck or bridge pickup, it works fine. When the switch is set for both, only the bridge pickup comes through. I can't figure out where the problem is, since both pickups work solo, but not when joined. BTW, I checked the switch for continuity, and the circuits in all three positions seem fine.
If the switch shows proper continuity in all positions and the pickups are definitely good it sounds like a ground issue. Check a schematic. If the schematic checks out, un solder the switch from the pickups, clean all solder off switch, snip the pickup wires clean, restrip and tin the wires, and resolder the pickups to the switch. If that doesn't work, you can double check the pickups by switching and rewiring them. If you have the same problem, it's the switch or the way it is wired. It's either the switch or the wiring assuming the pickups are good.