We formulate a generalized tunneling-percolation model that describes the electrical properties of conductor-insulator cellular disordered systems. This model predicts and explains the experimentally observed universal and nonuniversal behaviors of such properties and the crossover between them. Studying both the conductance and the corresponding piezoresistive response coefficient we show, by using effective-medium theories and Monte Carlo calculations, that the crossover of the conductivity-percolation exponent is reflected by a corresponding crossover in the behavior of the above coefficient, thus providing a more practical means to characterize the critical region of the system.