Near the end of the film when Frank Sr. is getting in his car to head for Europe, he tosses his hat on the seat with his coat on top of it. After saying goodbye to his family, he gets into the car and the hat is on top of the coat.
During the high-school dance, the band plays "Lucky Lindy," a 1929 song describing Charles A. Lindbergh's solo trans-Atlantic flight in 1927. Frank Gilbreth died in 1924--three years before Lindbergh's flight and five years before the song became popular.
In the opening shot of Providence, RI seen a railroad yard, clearly visible on the track is a diesel switch engine. The first diesel locomotives had not been invented until the mid 1930's. The movie is set in the mid 1920's.
As the camera zooms in during the tonsillectomy, you can see the shadow of the camera on the back of one of the nurses' white coats.
The shadow of the camera equipment is visible on the grass when Frank Sr. is telling everyone goodbye when he is leaving for Europe.
The shadow of the boom mic is visible in the dining room at the Nantucket summer home scene.
At a party, a man gets a dark purple drink from the punch bowl and tells a woman, "I hope you like pineapple."