The Stage Manager mentions the "Boston train" and the "Albany train" passing through Grover's Corners, NH. The Boston-Albany train line did not (and does not) enter New Hampshire.
The movie begins on June 7, 1901. That night everyone comments on how beautiful the moon is, and it is later learned that it is 8:30 p.m. The moon did not rise until midnight on that date. In fact, the sun did not set until 8:30 p.m. Additionally, crickets can be heard loudly chirping throughout these scenes. This would be unusual for early June, but common in late August.
Sunflowers are in full bloom in the garden. This wouldn't occur in early June, especially that far north in the U.S.
Studio lights are reflected on the mourners' rain-soaked umbrellas.
It is mentioned that Grover's Corners is found at 42 degrees 40 minutes latitude and 70 degrees 37 minutes longitude. These coordinates are actually off the coast of Boston, Massachusetts in the Atlantic Ocean, not inland near the Massachusetts/New Hampshire border.
Mr. Webb says that of the residents of Grover's Corners, 86% are Republican, 12% are Democrats, 4% are socialists - the rest "indifferent." That's a total of 102%.