perturbation theory - Feynman rules and the hydrogen atom

In his Quantum theory of fields, Vol. 1, Weinberg claims that the old-fashioned perturbation theory allows studying an appearance of singularities in matrix elements by intermediate states.

As an example, he considers the scattering of an electron on a proton and argues that the old-fashioned perturbation theory says that due to the intermediate state the true expansion parameter in the task is not the $e^{2}$, but $e^{2}m_{e}/q$, where $q$ is the momentum of the proton and the electron in the CM frame. Therefore, for $q\lesssim e^{2}m_{e}$ the perturbation theory breaks down, and we obtain a bound state - the hydrogen atom.

My question is the following: why after "rephrasing" in terms of Lorentz invariant quantities (i.e., in terms of Feynman rules) the perturbation theory loses its property to trace such singularities?