Ab initio electronic structure calculations employing the fixed-spin moment procedure and the generalized gradient correction are used to explore the mechanical stability of α-iron and related magnetic materials, including body-centered-cubic cobalt and ziconium. The mechanical stability is found to depend sensitively on the magnetization. At relatively low values of the magnetization, specific vibrational and elastic instabilities are identified and are interpreted in terms of the behavior of the electronic density of states at the Fermi level. These results also raise the possibility of influencing the stability of bulk magnetic phases by external fields.