Qualitatively incorrect results are obtained for the Mn dimer in density functional theory calculations using the generalized gradient approximation (GGA), and similar results are obtained from local density and meta-GGA functionals. The coupling is predicted to be ferromagnetic rather than antiferromagnetic, and the bond between the atoms is predicted to be an order of magnitude too strong and approximately an Ångstrøm too short. Explicit, self-interaction correction (SIC) applied to a commonly used GGA energy functional, however, provides close agreement with both experimental data and high-level, multireference wave function calculations. These results show that the failure is not due to a strong correlation but rather the single electron self-interaction that is necessarily introduced in estimates of the classical Coulomb and exchange-correlation energy when only the total electron density is used as the input. The corrected functional depends explicitly on the orbital densities and can, therefore, avoid the introduction of a self-Coulomb interaction. The error arises because of an overstabilization of bonding d-states in the minority spin channel resulting from an overestimate of the d-electron self-interaction in the semilocal exchange-correlation functionals. Since the computational effort in the SIC calculations scales with the system size in the same way as for regular semilocal functional calculations, this approach provides a way to calculate properties of Mn nanoclusters as well as biomolecules and extended solids, where Mn dimers and larger cluster are present, while multireference wave function calculations can only be applied to small systems.