A new characterization of nonradiating (NR) sources is derived that is based on electromagnetic potentials. In the new description a hierarchy of NR sources is systematically created that includes certain nonlocalized NR sources having the property that their curl is localized. The important class of spatially localized NR sources, whose fields vanish everywhere in the exterior of the source, corresponds to a special case of the general theory. The new NR source developments are discussed in connection with the question of measurability of electromagnetic potentials as enabled by the Aharonov-Bohm (A-B) effect, whereby quantum mechanical effects of the potentials can be observed in regions of vanishing electromagnetic fields but nonvanishing electromagnetic potentials. A necessary condition is derived for an electrodynamic A-B effect in the exterior of a spatially localized NR source. By exploring this condition, it is concluded that for time-varying, information-carrying fields (as required, e.g., in communications and remote sensing applications) the required A-B conditions of vanishing fields and nonvanishing potentials are not possible in the exterior of a NR source; i.e., electrodynamically, if the fields vanish everywhere outside the source, then the potentials also vanish there. This does not necessarily hold under static conditions in which nontrivial potentials with physically observable quantum effects can exist in the exterior of a source having zero external fields.