Multiple times during the final (Ingrid Bergman) story you hear the sound effect of the car tires screeching on a dirt/gravel road. This sound happens only on asphalt or concrete, never on a low-traction surface like dirt.
As the Rolls leaves the race track there is no wreath on the front but there is a wreath on the front as it pulls up to the mansion but it doesn't stop once on the way there to have the wreath put on the car.
When Shirley MacClaine swims into the Blue Grotto, her hair is wet and stringy, but once inside the grotto, while the rest of her body is still wet, her hair is dry and neatly coiffed.
Art Carney and Shirley MacLaine are sitting at an outdoor cafe after George C. Scott returns to the U.S., and a waiter brings them each a drink in a different type of glass without either one of them seeing or giving their order to the waiter first.
In the opening titles, the roofs of modern cars can be seen as the camera pans along Hyde Park.
In the last scene, supposedly in the 1940s, the Rolls is driving north on the Henry Hudson Parkway passing under an overhead sign for Interstate 95. The Interstate System was not conceived of until the Eisenhower administration and I-95 was not designated through Manhattan until 1957.
Although the first two 'episodes' of the film are set prior to 1941, Shirley MacLaine's wigs and costumes are all contemporary to 1964.