The nonlinear mechanical properties of cells in response to large forces as they arise under physiological conditions are not well understood. In this work, a magnetic tweezers setup optimized for large forces was developed. Forces of more than 100 nN can be applied, which has not been achieved previously with a setup of this type. The time- and force-dependent nonlinear material properties of various cell types were examined using this setup. The creep response always followed a power law, as in the linear case. With increasing force, stress stiffening and shear fluidization were observed. The force dependence of these two parameters reveals a simple relationship: the differential stiffness is proportional to the sum of internal and external mechanical stress of the cytoskeleton. Consequently, cells control not only their linear, but also their nonlinear mechanical properties via their active motor-protein generated internal prestress. In the third part of this work, a comprehensive model of cell mechanics was developed. Numerical simulations show that the model qualitatively reproduces the experimental observations.