What sort of worm and worm-gear are you using? It is possible to make reversible worm drives - though they usually have two- or three-start threads.
Even with a worm gear that cannot be reversed if you are causing a sudden impact on the suspended plate it may result in sufficient short term force with bounce to joggle the motor shaft.
You say "then the motor should disengage." Does that mean that the motor is physically disconnected from the shaft with the worm? If so the opportunity for joggling would be much greater. If, however, you just mean that power to the motor is switched off then why not say so.
If the only purpose of the motor is to adjust the angle of the plate then a stepper motor might be more appropriate as it will actively hold its position in the face of attempts by the worm gear to joggle it - assuming, of course, that the stepper is permanently connected to the worm shaft and the stepper is properly powered all the time.
...R
PS ... the more I think about this
where the motor can rotate freely despite not being supplied power, but attempting to rotate the plate outside of that range will not work
the more I suspect that the problem is caused by the load "bouncing" so that it alternately loads and unloads the interface between the worm gear and the worm. Most systems that rely on a worm holding position have a constant load - think of a winch. Those systems also probably aren't concerned with the same precision as you are.
PPS ... a possible solution may be to use a servo to apply a brake to the motor shaft during the tests