From The Making of Casablanca: Bogart, Bergman, and World War II:
Casablanca started on Stage 12A with the flashback to Rick and Ilsa’s romance in Paris. It was an accident that Bogart was required to make love to Bergman almost before he was introduced to her. Originally, production was to start in Rick’s Cafe on Stage 8, but the intricate clockwork that matched actors, scripts, stages, and sets had been thrown off because Irving Rapper was two weeks behind schedule on Now, Voyager. Claude Rains didn’t finish his role as the wise psychiatrist in Now, Voyager until June 3. Paul Henreid was not free until June 25. So the [Michael] Curtiz movie began with the scene in the Montmartre cafe. The first day, a lovestruck Richard Blaine — “His manner is wry but not the bitter wryness we have seen in Casablanca” say the stage directions — pours champagne for himself, Ilsa, and Sam while the Germans march toward Paris and Sam plays, “As Time Goes By”.
According to Geraldine Fitzgerald, Bogart and Bergman had lunch together a week or ten days before Casablanca started production. “I had lunch with them,” she says. “And the whole subject at lunch was how they could get out of the movie. They thought the dialogue was ridiculous and the situations were unbelievable. And Ingrid was terribly upset because she said she had to portray the most beautiful woman in Europe, and no one would ever believe that. It was curious how upset she was by it. ‘I look like a milkmaid,’ she said.

