When linear Pade summation is applied to eigenvalue perturbation expansions near regions of parameter space where those eigenvalues undergo an avoided crossing, the Padi approximants may yield levels which cross diabatically, rather than displaying the proper avoided behavior. The purpose of this study is to elucidate the reasons for the peculiar behavior of Pade approximants in such situations. In particular : we demonstrate that the diabatic crossing is a natural consequence of using the (single-valued) Pade rational approximant to successfully resum series expansions of the multivalued energy function over much of the parameter space. This is illustrated with a perturbative treatment of the Barbanis Hamiltonian.