Density functional theory (DFT) is widely applied in calculations of molecules and materials. Yet, it suffers from a well-known over-emphasis on charge delocalization arising from self-interaction error that destabilizes localized states. Here, using the symmetric diamine N,N'-dimethylpiperazine as a model, we have experimentally determined the relative energy of a state with positive charge localized on one of the two nitrogen atoms, and a state with positive charge delocalized over both nitrogen atoms. The charge-localized state was found to be 0.33 (0.04) eV higher in energy than the charge-delocalized state. This provides an important test of theoretical approaches to electronic structure calculations. Calculations with all DFT functionals commonly used today, including hybrid functionals with exact exchange, fail to predict a stable charge-localized state. However, the application of an explicit self-interaction correction to a semi-local functional identifies both states and gives relative energy in excellent agreement with both experiment and CCSD(T) calculations.