The right-handed neutrinos within the type-I seesaw mechanism can induce large radiative corrections to the Higgs mass, and naturalness arguments can then be used to set limits on their mass scale and Yukawa couplings. Driven by minimality, we consider the presence of two degenerate right-handed neutrinos. We compare the limits from naturalness with the ones from the stability of the electroweak vacuum and from lepton flavor violation. Implications from neutrinoless double beta decay are also discussed and renormalization effects for the light neutrino parameters are presented. Adding small perturbations to the degenerate heavy neutrino spectrum allows for successful leptogenesis. © 2017 American Physical Society.