We demonstrate that the dispersion-corrected density functional theory (DFT-D) schemes fall short of predicting reliable three-body interaction energies. This concerns also a popular variant of DFT-D called the "many-body dispersion" (MBD) method, which might seem surprising in the light of the fact that its name contains the very phrase "many-body". The main reason for the inaccuracy of the three-body interaction energies in the DFT-D schemes can be attributed to internal deficiencies of the standard DFT functionals that the existing "-D" methods are incapable of correcting since the main problems emerge from the terms not related to the dispersion component. At present, it seems that none of the a posteriori dispersion techniques are able to predict accurately the total interaction energy for a supermolecular system together with its simultaneous decomposition into the many-body components. On the other hand, if one is interested only in the three-body interaction energies, we propose an adjustment to the MBD approach that achieves good accuracy in conjunction with the supermolecular MP2.