It is commonly accepted that the addition of spherical nanoparticles (NPs) cannot simultaneously improve the elastic modulus, the yield stress, and the ductility of an amorphous glassy polymer matrix. In contrast to this conventional wisdom, we show that ductility can be substantially increased, while maintaining gains in the elastic modulus and yield stress, in glassy nanocomposite films composed of spherical silica NPs grafted with polystyrene (PS) chains in a PS matrix. The key to these improvements are (i) uniform NP spatial dispersion and (ii) strong interfacial binding between NPs and the matrix, by making the grafted chains sufficiently long relative to the matrix. Strikingly, the optimal conditions for the mechanical reinforcement of the same nanocomposite material in the melt state is completely different, requiring the presence of spatially extended NP clusters. Evidently, NP spatial dispersions that optimize material properties are crucially sensitive to the state (melt versus glass) of the polymeric material.