We have studied the nucleation of crystals of a model protein from aqueous solutions using a never technique that allows direct determinations of homogeneous nucleation rates. At a constant temperature of 12.6 degrees C we varied the thermodynamic supersaturation by changing the concentrations of protein and precipitant. We found a broken dependence of the homogeneous nucleation rate on supersaturation that is beyond the predictions of the classical nucleation theory. The nucleation theorem allows us to relate this to discrete changes of the size of the crystal nuclei with increasing supersaturation as (10 or 11) -> (4 or 5) -> (1 or 2). Furthermore, we observe that the existence of a second liquid phase at high protein concentrations strongly affects crystal nucleation kinetics: (i) Crystal nucleation rates are lower than expected in the phase region of liquid-liquid demixing. (ii) In the immediate proximity of this region, nucleation rates vary by factors of up to 2 in identical experiments. Since for this region theory predicts a sharp rate maximum, we attribute this kinetic instability to minor shifts of the experimental conditions toward or away from the phase boundary.