The dataset contains the experimental dynamic measurements of a nonlinear Multi-Degree of Freedom System featuring a hardening stiffness characteristic. The system was studied in two different ways: the response associated with the underlying linear system was measured by exciting the test rig with low-amplitude random excitation while the full-nonlinear response of the system was measured by exciting the test rig via stepped sine excitation at different excitation amplitudes. All the data are stored in MAT and CSV files which are accessible with MATLAB codes (attached to the dataset).