The semiclassical description of Skyrmions at small isospin chemical potential μ_I is carefully analyzed. We show that when the calculation of the energy of a nucleon is performed using the straightforward generalization of the vacuum sector techniques (μ_I=0), together with the “natural” assumption μ_I=O(N_c⁰), the proton and neutron masses are nonlinear in μ_I in the regime |μ_I|<m_π. Although these nonlinearities turn out to be numerically quite small, such a result fails to strictly agree with the very robust prediction that for those values of μ_I the energy excitations above the vacuum are linear in μ_I. The resolution of this paradox is achieved by studying the realization of the large N_c limit of QCD in the Skyrme model at finite μ_I. This is done in a simplified context devoid of the technical complications present in the Skyrme model but which fully displays the general scaling behavior with N_c. The analysis shows that the paradoxical result appears as a symptom of using the semiclassical approach beyond its regime of validity and that, at a formal level, the standard methods for dealing with the Skyrme model are only strictly justified for states of high isospin, I~N_c.