When Mr. Rogers departs from the train at the Paris station; a photographer takes his photo as he shields his daughter. But on the next cut when the photo appears on the newspaper's front page; the composition of the photo is now different than what actually took place.
During "Old Piano Roll Blues," Fernando Lamas's hand movements make it obvious that he's not actually playing the piano.
Jim and Marie were legally married therefore the mother's name would have been on their daughter Elizabeth's birth certificate even despite the fact that Elizabeth was told that her mother died. Although Marie abandoned her marriage and her daughter; she did not change her name. It, therefore, doesn't seem plausible that the now grown-up Elizabeth would not know her mother's name and not become somewhat suspicious upon meeting Marie while in Paris. This story gap was not addressed in the film.
When Andre arrives inside the club, he is not aware that Elizabeth is also there. However, since her father Jim was waiting for her at the club's entrance; Jim and Andre would have seen each other therefore Andre would have known that Elizabeth is in the club. The film doesn't offer an explanation of how Andre and Jim could have possibly managed to avoid seeing each other.