A recent idea, put forward by Mund, Rehren and Schroer, is discussed; it suggests that in gauge quantum field theory one can replace the point-localized gauge fields by gauge-invariant, though string-localized vector potentials and a principle of string-independence. Based on a kinematical model, describing unmovable (static) point particles carrying opposite charges, it is shown that these string-localized potentials cannot be used for the description of the gauge bridges between electric charges. These bridges are needed in order to ensure the validity of Gauss' law. This observation does not preclude the existence of Poincar\'e invariant theories, describing the coupling of string-localized gauge invariant potentials to matter fields. But such theories can hardly be equivalent to `usual' quantum electrodynamics, as hoped for by their proposers.