State of the art ab initio calculations in a fully relaxed geometry reveal the interplay between structure and magnetism in atomic-sized nanocontacts. Our studies for Co, Pd, and Rh nanocontacts sandwiched between Cu electrodes demonstrate that atomic relaxations strongly affect magnetic states and lead to an inhomogeneous distribution of magnetic moments on atoms of nanocontacts. Stable ferromagnetic solutions with large magnetic moments are found for Co and Rh nanocontacts before breaking of the contact. We predict that Pd nanocontacts are nanomagnetic before the breaking, however, the energy difference between ferromagnetic and nonmagnetic states is only 6 meV. Our results suggest that variations in the structure, temperature or applied field could lead to transitions between magnetic and nonmagnetic states in Pd nanocontacts.