To first order, gravity is indeed linear in the way you suggest, so you can just add up the effects of different planets separately. To see why this is the cas

To first order, gravity is indeed linear in the way you suggest, so you can just add up the effects of different planets separately. To see why this is the case, note that the Sun itself could be considered to be a collection of independent masses. If there were some non-linear gravitational effect, then the gravity of the separate components would be different from the gravity of them all together. However, the surface gravity can be measured across a wide range of masses and scales as you would predict from Newtonian gravity. That being said, general relativity predicts that there is in fact a non-linear scaling when gravitational fields are extremely strong. The reason for this is that the there is some energy associated with the gravitational field, and any energy produces its own gravitational field. So the gravitational field itself exerts its own gravitational field, leading to the non-linearity.