The stability of large magnetization motions in systems with uniaxial symmetry subject to a circularly polarized radio-frequency field is analytically studied. Instability conditions valid for arbitrary values of the amplitude and frequency of the driving field are derived. In the limit of small motions, these conditions yield Suhl's theory of spin-wave instabilities for the case of ferromagnetic resonance. It is shown that the input powers capable of inducing spin-wave instabilities are bounded from both below and above, so that large enough motions are always stable. In addition, it is demonstrated that stability of uniform motions depends on their preparation history.