Many (most) integrated regulators don't like being backdriven.
***the reverse protection diode would take care of that - but only if each regulator had its own diode, and in any case I agreed with whoever said a single 5V reg would do. ***Phil already pointed out the forward conduction diode knocks 0.6V off the output - If the OP really does feel the need to include that diode, then the correct output can be restored by grounding the reference pin via a compensating diode with a bias resistor to force a forward volt drop.