Anharmonic vibrational calculations for the benzoic acid monomer and dimer in the mid-IR regime (500...1800 cm^-1) are reported. Harmonic frequencies and intensities are obtained at the DFT/B3LYP level of theory employing D95(d,p) and cc-pVTZ basis sets. Anharmonic corrections obtained from standard perturbation theory lead to red shifts of 1...3 %. In almost all cases, the resulting frequencies deviate by less than 1 % from previous measurements [Bakker et al., J. Chem. Phys. 119, 11180 (2003)]. Calculated intensities are in qualitative agreement with the absorption experiment, with the cc-pVTZ values being superior to the D95(d,p) ones for a few modes of the dimer. The antisymmetric out-of-plane bending mode of the dimer, which is strongly blue-shifted with respect to the monomer frequency, represents a remarkable exception: The harmonic frequencies obtained for the two basis sets differ notably from each other, and the anharmonically corrected frequencies deviate from the experimental value by 8 % (D95(d,p)) or 3 % (cc-pVTZ). Non-perturbative calculations in reduced dimensionality reveal that the relatively small total anharmonic shift (few tens of cm^-1) comprises of partly much larger contributions (few hundreds of cm^-1) which are mostly canceling each other. Many of the individual anharmonic couplings are beyond the validity of second order perturbation theory based on cubic and semi-diagonal quartic force constants only. This emphasizes the need for high-dimensional, non-perturbative anharmonic calculations at high quantum chemical level when accurate frequencies of H-atom vibrations in double hydrogen bonds are sought for.