Non-covalent interactions determine in large part the thermodynamic aspects of molecular packing in organic crystals. Using a combination of symmetry-adapted perturbation theory (SAPT) and classical multipole electrostatics, we describe the interaction potential energy surfaces for dimers of the oligoacene family, from benzene to hexacene. An analysis of these surfaces and a thorough assessment of dimers extracted from the reported crystal structures underline that high-order interactions (i.e., three-body non-additive interactions) must be considered in order to rationalize the details of the crystal structures. A comparison of the SAPT electrostatic energy with the multipole interaction energy demonstrates the importance of the contribution of charge penetration, which is shown to account for up to 50% of the total interaction energy in dimers extracted from the experimental single crystals; in the case of the most stable co-facial model dimers, this contribution is even larger than the total interaction energy. Our results highlight the importance of taking account of charge penetration in studies of the larger oligoacenes.