The recent discovery of a superconducting transition at 39 K in MgB₂—made of alternating Mg and graphene-like B planes—has raised great interest, for both its technological and theoretical implications. It was clear since the very beginning that the properties of this material are related to an anomalous coupling between the charge carriers in the σ bands—due to in-plane bonds between Boron atoms—and the phonon mode (E_2g) which involves in-plane vibrations of the B ions. Theoretical studies have thus been focused on the search for possible anomalies in the e–ph coupling: one of the first results was the discovery that the E_2g phonon is highly anharmonic, but the connection between anharmonicity and T_c in this material is still a controversial point. We first present a detailed first-principles study of the E_2g phonon anharmonicity in MgB₂ and analogous compounds which are not superconducting, AlB₂ and graphite, and in a hypothetical hole-doped graphite (C²⁺₂); we then introduce an analytical model which allows us to relate the onset of anharmonicity with the small Fermi energy of the carriers in σ bands. Our study suggests a possible relation between anharmonicity and non-adiabaticity; non-adiabatic effects, which can lead to a sensible increase of T_c with respect to values predicted by conventional theory, become in fact relevant when phonon frequencies are comparable to electronic energy scales.