Hi heartlessphil,
according to current documentation it uses a GLOBAL clip plane, hence ONE for everything.
You can have multiple Planer Reflection Actors, but if you do they need to face the same direction AND need to be on the same height level, otherwise the reflections clip planes might be off.
If you orient them differently the exact artifacts you described may happen. 
Please note that most of this is based interpretation of what is described in documentation here: https://docs.unrealengine.com/latest/INT/Engine/Rendering/LightingAndShadows/PlanarReflections/
Have fun