We investigate the relation between the structure and the viscoelastic behavior of a model polymer nanocomposite system based on a mixture of titanium dioxide (TiO2) nanoparticles and polypropylene. Above a critical volume fraction, Phi(c), the elasticity of the hybrids dramatically increases, and the frequency dependence of the elastic and viscous moduli reflects the superposition of the independent responses of the suspending polymer melt and of an elastic particle network. In addition, the elasticity of the hybrids shows critical behavior around Phi(c). We interpret these observations by hypothesizing the formation of a transient network, which forms due to crowding of particle clusters. Consistent with this interpretation, we find a long-time, Phi-dependent, structural relaxation, which emphasizes the transient character of the structure formed by the particle clusters. For times below this characteristic relaxation time, the elasticity of the network is Phi-independent and reminiscent of glassy behavior, with the elastic modulus, G', scaling with frequency, omega, as G' similar to omega(0.3). We expect that our analysis will be useful for understanding the behavior of other complex fluids where the elasticity of the components could be superimposed.