The origin of non-additivity in hydrogen bonds (H-bonds), usually termed as H-bond cooperativity, is investigated in H-bonded linear chains. It is shown that H-bond cooperativity originates solely from classical electrostatics. The latter is corroborated by comparing the H-bond cooperativity in infinitely-long H-bonded hydrogen cyanide, 4-pyridone and formamide chains, assessed using density functional theory (DFT), against the strengthening of the dipole-dipole interaction upon the formation of an infinite chain of effective point-dipoles. It is found that the magnitude of these effective point-dipoles is a consequence of mutual polarization and additional effects beyond a polarizable point-dipole model. Nevertheless, the effective point-dipoles are fully determined once a single H-bond is formed, indicating that quantum effects involved in H-bonding are circumscribed to nearest-neighbor interactions only; i.e. in a linear chain of H-bonds, quantum effects do not contribute to the H-bond non-additivity. This finding is verified by estimating cooperativity along the dissociation path of H-bonds in the infinite chains, using two empirical parameters that account for polarizability, together with DFT association energies and molecular dipoles of solely monomers and dimers.