Active masses of solids and liquids
You might view the relevant principle as follows: components whose activity changes during a reaction tend to put a stop to a reaction before they are completely consumed. Components with constant activity (independent of the amount of other reactants), on the other hand, can be completely consumed (exhausted): they can be limiting reagents, provided they are less abundant (stoichiometrically) than the other reagents. If a reagent with constant activity is exhausted before equilibrium is reached the reaction stops - it is then a limiting reagent. Reagents whose activities change during a reaction (because their concentration changes, for instance) are in principle not exhausted this way because their activity increases too much for the reaction to proceed any further. Reactions involving only reagents with variable activity tend to go to an equilibrium in which none of the species is completely exhausted. If reagents with constant activity are present in excess (such that their initial amount exceeds that at equilibrium), then they will not limit the extent of reaction. Note also that if the increase in activity with dilution of a reagent with variable activity is negligible, its concentration at equilibrium may be so low that it can be considered exhausted (and limiting). 

A good example of the relevant principle is a mixture of two reactive solids or two immiscible liquids of constant activity that react at the interface between them. In either case, if the reaction is favorable ($Q\gt K_{eq}$) then it will proceed until one or both of the substances has been completely consumed (both being completely consumed if present in stoichiometric proportion). On the other hand, when a solid dissolves into a solvent, the activity of the solid remains constant and it will continue to dissolve provided the activity of the dissolved species does not increase above that of the solid. If it doesn't, then the solid is completely dissolved (exhausted).

If a reaction is already at equilibrium, adding more of a solid won't make a difference (provided there are no surface effects). But if you are not at equilibrium, it will if the amount present is limiting. A good example again is solubilization of a solid in a solvent. Added solid will continue dissolving until you reach the saturation point. Past this point the concentration of the compound in solution does not change any further. In addition, with saturation the activity of the compound in the solid and solution phases becomes equal.