The cumulant expansion has become a popular tool for describing the dynamics of electrons coupled to phonons. Here, using the Holstein model, the authors study the behavior of low-order cumulants, and present a self-consistent formulation of the cumulant expansion. They find that the second-order cumulant performs well at zero electronic momentum or high temperature, and the fourth order cumulant becomes pathological, preventing its general application. The self-consistent formulation greatly improves the spectra by including non-perturbative effects, but also presents pathologies. These results have implications for calculations of transport properties and systems.