The story is set in the years 1873 - 1888, yet Billy uses the term "midway" while describing the scene around the carousel. This definition of "midway", referring to the area of a circus or carnival containing sideshows and other amusements, was coined during the Chicago Columbian Exposition of 1893 to refer to the area of tawdry exhibits located on a narrow strip of land connecting Jackson and Washington Parks which was called "Midway Plaisance". The term "midway" would not have been used in reference to a carnival or circus prior to 1893.
The story supposedly begins in the year 1873. But the carousel is festooned with numerous colorful incandescent light bulbs. Thomas Edison did not introduce the first practical incandescent light bulb until 1879.
Mockingbird calls are heard in a scene between Billy and Louise near the end. At the time the film is set (between the early 1870s and the late 1880s), the mockingbird range was entirely in the American South. There were no mockingbirds in Maine - not even in the year the film was made (by which time the mockingbird's range had extended to New York and New Jersey).
In the Starkeeper's office, after Billy has died, you can see the shadow of the boom mic on the Starkeeper's chair. It is right at the point when the Starkeeper says, "You couldn't bear to see her cry."
During the the song "June Is Busting Out.." the length of the shadows (presumably around mid-day) suggest either A) "January Is Busting Out All Over" or B) It's June in Nome, Alaska instead of New England.