R2NiMnO6 (R = Tb, Ho, Er, Tm) perovskites have been prepared by soft-chemistry techniques followed by high oxygen-pressure treatments; they have been investigated by X-ray diffraction, neutron powder diffraction (NPD), and magnetic measurements. In all cases the crystal structure is defined in the monoclinic P2(1)/n space group, with an almost complete order between Ni2+ and Mn4+ cations in the octahedral perovskite sublattice. The low temperature NPD data and the macroscopic magnetic measurements indicate that all the compounds are ferrimagnetic, with a net magnetic moment different from zero and a distinct alignment of Ni and Mn spins depending on the nature of the rare-earth cation. The magnetic structures are different from the one previously reported for La2NiMnO6, with a ferromagnetic structure involving Mn4+ and Ni2+ moments. This spin alignment can be rationalized taking into account the Goodenough-Kanamori rules. The magnetic ordering temperature (T-CM) decreases abruptly as the size of the rare earth decreases, since T-CM is mainly influenced by the superexchange interaction between Ni2+ and Mn4+ (Ni2+-O-Mn4+ angle) and this angle decreases with the rare-earth size. The rare-earth magnetic moments participate in the magnetic structures immediately below T-CM.