A quadratic configuration interaction calculation of the structure and magnetic hyperfine splitting constants in the tetranitrogen radical cation is reported. The species is determined to be linear, and splittings in close accord with experimental findings are calculated despite the large amount of spin contamination in the unrestricted Hartree-Fock (UHF) reference wave function. Indeed, reasonable agreement is attained even with a modest double-zeta description of the single-particle space in the molecular orbital treatment, provided that the effect of amplitudes due to single replacements in the reference wave function is included in the cluster expansion. Improvements in the quality of the basis set lead to improved predictions of the unpaired spin density, indicating that such agreement is not fortuitous. An analysis is given of the contributions from various levels of excitation, up to quadruples, in the UHF determinant. Analogous results are also presented for various ionic states of atomic nitrogen and the dinitrogen radical cation.