The problem of theoretical explanation of the experimentally observed linear stiffening of living cells is addressed. This explanation is based on Ingber's assumption that the cell cytoskeleton, which enjoys tensegrity architecture with compressed microtubules that provide tension to the microfilaments, affects the mechanical behavior of the living cell. Moreover, it is shown that the consideration of the extreme flexibility of microtubules and the unilateral response of microfilaments is crucial for the understanding of the living cell overall behavior. Formal nonlinear structural analysis of the cell cytoskeleton under external mechanical loads is performed. For this purpose, a general computer model for tensegrity assemblies with unilateral microfilaments and buckled microtubules is developed and applied to the theoretical analysis of the mechanical response of 2D and 3D examples of tensegrity cells mimicking the behavior of real living cells. Results of the computer simulations explain the experimentally observed cell stiffening. Moreover, the theoretical results predict the possible existence of a transient softening behavior of cells, a phenomenon, which has not been observed in experiments yet.