The membranes of human red blood cells (RBCs) are a composite of a fluid lipid bilayer and a triangular network of semiflexible filaments (spectrin). We perform cellular microrheology using the dynamic membrane fluctuations of the RBCs to extract the elastic moduli of this composite membrane. By applying known osmotic stresses, we measure the changes in the elastic constants under imposed strain and thereby determine the nonlinear elastic properties of the membrane. We find that the elastic nonlinearities of the shear modulus in tensed RBC membranes can be well understood in terms of a simple wormlike chain model. Our results show that the elasticity of the spectrin network can mostly account for the area compression modulus at physiological osmolality, suggesting that the lipid bilayer has significant excess area. As the cell swells, the elastic contribution from the now tensed lipid membrane becomes dominant.