The electronic structure of disordered systems is analyzed in the case of non-interacting electrons tightly bound to their ions. Particular attention is paid to the behaviour of the band gap in the presence of disorder. The Kronig-Penney and tight-binding solutions are derived for perfect lattices. With the physically reasonable restriction that the ions do not approach closer to each other than a certain distance, the band gap is discussed in successively more difficult cases. The closure of the gap between neighbouring levels in one dimension depends on the relative parity of the two levels: if the parities are the same the gap cannot close. The hardest case discussed, that of neighbouring s-levels in three dimensions, gives the same result. A detailed discussion of the approximations is given.