The ingenious Tonomura et al. experiments testing the existence of the Aharonov-Bohm (AB) effect is analyzed. It turns out that they can be described with both singlevalued and multivalued wave functions. Thus, the Tonomura experiments neither confirm nor reject the AB effect existence. A slight modification of these experiments is proposed which can resolve the above ambiguity and elucidate the possible used of multivalued wave functions in multiconnected space regions. 10 refs.; 2 figs.