In this work we investigate the nonlinear dynamical interaction of magnetized electrons and ordinarily polarized electromagnetic modes. We show that when either the mode amplitude is sufficiently large or the electronic longitudinal momentum along the external magnetic field is sufficiently small, non-integer harmonics originating from higher-order terms in the expansion of the Hamiltoniari in powers of the electromagnetic field become important, and the relevant action-angle phase space ceases to be well described by standard nonlinear pendulum approximations. We develop appropriate resonance overlap studies in order to determine the conditions to be satisfied at the transition from regular to chaotic orbits, checking the accuracy of the results with numerical integrations of the full dynamical equations. We verify that chaotic orbits occur within a ‘window’ among the possible values of the longitudinal momentum. The width of this window and the degree of overlap between neighbouring resonances within it are both shown to grow with particle action and wave frequency. We also show that once overlap has occurred for a particular value of the action, it persists for larger values of this quantity, a circumstance that leads to a large amount of particle energization.