We employ experiment and theory to explore the nonlinear elasticity and yielding of concentrated suspensions of nanoparticles which interact via purely repulsive forces. These glassy suspensions are found to exhibit high exponent power law or simple exponential dependences of the shear elastic modulus and perturbative yield stress on nanoparticle volume fraction, as well as a monotonic decrease of the perturbative yield strain with increasing concentration. Our experimental observations are in good agreement with the predictions of a recently developed microscopic statistical mechanical theory, which describes glassy dynamics based on a nonequilibrium free energy that incorporates local cage correlations and activated barrier hopping processes [(1) Schweizer, K. S.; Saltzman, E. J. J. Chern. Phys. 2003, 119, 1181. (2) Saltzman, E. J.; Schweizer, K. S. J. Chem. Phys. 2003, 119, 1197. (3) Kobelev, V. Schweizer. K. S. Phy. Rev. E 2005, 71, 021401].(1-3)