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Within the framework of Bonnor's exact solution describing a massive magnetic dipole, we study the motion of neutral and electrically charged test particles. In dependence on the Bonnor spacetime parameters, we determine regions enabling the existence of stable circular orbits confined to the equatorial plane and of those levitating above the equatorial plane. Constructing Poincare surfaces of section and recurrence plots, we also investigate the dynamics of particles moving along general off-equatorial trajectories bound in effective potential wells forming around the stable circular orbits. We demonstrate that the motion in the Bonnor spacetime is not integrable. This extends previous investigations of generalized Stormer's problem into the realm of exact solutions of Einstein-Maxwell equations, where the gravitational and electromagnetic effects play a comparable role on the particle motion.