A strongly nonlinear rotor-bearing system often has multiple solutions under harmonic excitations and jump phenomena. For example, a hardening nonlinearity may include a jump-down in the acceleration process and jump-up in the deceleration process. It is challenging to measure all of these multiple responses and establish an accurate dynamic model from experimental data to predict these phenomena. This paper used a fixed frequency test method to measure all of these multiple responses under harmonic excitations and developed a novel strategy to characterize and identify nonlinearities in a strongly nonlinear rotor-bearing system based on reconstructing constant response tests from fixed frequency test data. The fixed frequency tests are achieved by monotonically increasing the voltage applied to the exciter at a fixed frequency and using the force drop-out phenomenon through the resonance to control the force applied to the structure. This test method could measure multivalued response curves of a strongly nonlinear rotor-bearing system in a nonrotating state. The constant response tests could be reconstructed from these multivalued response curves. The relationship of equivalent stiffness versus displacement can be established, and hence, the nonlinear stiffness is characterized and identified from constant response tests. A rotor-bearing system with a strongly nonlinear support is used to demonstrate the method, and the nonlinear support stiffness parameters are identified and validated in a nonrotating state. The identified nonlinear rotor-bearing model also could predict the jump phenomena in the acceleration or deceleration process. The results demonstrate the feasibility and effectiveness of the approach, and also show the potential for practical applications in engineering.