The nonlinear frequency responses of a spin-wave (magnonic) active ring resonator have been studied in a wide range of bias magnetic fields. The resonator consists of a magnetic thin film strip serving as a waveguide structure for nonlinear surface spin waves that are fed back through a variable gain microwave amplifier. An increase in the input power or amplifier gain coefficient causes a negative nonlinear shift of the eigen frequencies of the resonant modes of the ring. When the input power or amplifier gain coefficient exceeds some threshold value, we see appearance of bistability in the resonator behavior. The bistability manifests itself as a stable hysteretic response of the resonator in the form of two different values of the output power for the same operating frequency. The output power transits from low to high (from high to low) state when the frequency of the input signal is swept-up (swept-down). The frequency positions of the output power steps define the bistability frequency range. We observe broadening of the bistability frequency range with an increase in the bias magnetic field. In order to explain the experimental data, we developed a theory of the bistable behavior of the magnonic active ring resonators. The theory takes into account nonlinear phase shift and nonlinear damping of spin waves. It is in a good agreement with the experimental data and explains well the obtained experimental results.