Another possibility:
NOTE: I AM NOT familiar with the innards of PRS guitars. However, I do know Gibson guitars inside-out, and you mention a 3-way switch, so I'm assuming there is some similarity.
Gibson 3-way switches have 2 sets of contacts.
Assume the toggle switch is in the middle (both pickups) position. At this point, the switch is not engaging either set of contacts. Since both sets are normally-closed, the hot wires from both pickups get connected through their respective set of contacts.
Now flip the toggle down. The rhythm (neck) pickup is connected to the upper set of contacts. When you flip the toggle down, it engages the upper contacts, and opens the connection. That causes the neck pickup to be disconnected, while the treble pickups is still connected through its set of closed contacts.
Gabe probably has called out the right error. However, if your toggle isn't moving fully to one side (or the other), it might not fully disengage the closed contacts. That would allow some amount of bleedthrough (or a fully on pickup that should be off). Other causes could be solder bridges that short or partially-short the contacts, which then prevents the pickup from ever being shut off properly.
I've even seen too-fat toggle switch caps prevent the toggle from moving all the way to either top/bottom, and keep the guitar in "both-pickups" mode. The cure was generally to use a different switch tip. Rarely (but especially of the switch is replaced with a different type), the cavity the switch sits in is too narrow for the closed contacts to be opened fully. That requires a little bit of routing/sanding to enlarge the hole enough for the switch to move freely.