We report the observation of self-oscillations of the output power in a cascaded Raman fiber laser delivering two Stokes components. Our cascaded Raman fiber laser is made with a highly nonlinear photonic crystal fiber and it oscillates within a Perot-Fabry cavity formed by weak Fresnel reflections from the fiber ends. From our experimental and theoretical study, we identify stimulated Raman scattering as being the physical effect dominant in the emergence of unstable behaviors inside the Perot-Fabry cavity. Mechanisms of laser destabilization are thus found to be very different from polarization mechanisms previously identified as being responsible for unstable behaviors in conventional one-stage Raman fiber lasers [14, 15].