To better understand the contribution of cosmic ray muons to the CUORICINO background, 10 plastic scintillator detectors were installed at the CUORICINO site and operated during the final 3 months of the experiment. From these measurements, an upper limit of 0.0021 counts/(keV kg yr) (95% CL) was obtained on the cosmic ray-induced background in the neutrinoless double beta decay region of interest. The measurements were also compared to Geant4 simulations.