Revealing mistake: As the passengers are getting on the train at the start, their shadows reveal that the train car wheels and springs are painted on a flat surface. (00:00:10)
Revealing mistake: At a certain point we see the arid, dry Santa Monica Mountains of Los Angeles in the upper left of the picture with a huge high-tension electrical tower at the summit. The film is set in 1912 in Iowa, a green, rainy state which is also completely flat, with no mountains. Then again, we see the arid, dry Santa Monica Mountains in the upper right. (00:08:27)
Revealing mistake: During "Marian the Librarian," Marian is seen doing a little dance on the table before being hushed by the people in the library. Like the incident with Winthrop in "Gary, Indiana," you can see that she did not actually do this dance. Tap sounds were added in later to make it seem like she did.
Revealing mistake: At the end of Winthrop's version of "Gary, Indiana," we see what appears to be a little tap dance done by Winthrop that concludes the song. Upon close observation, however, you can tell that Winthrop does not actually do this dance. He just twirls around and tap noises are added in to make it seem like he is tapping.
Revealing mistake: During the July 4th festivities at Madison Park, Professor Harold Hill prompts four school board members (the Buffalo Bills quartet) into performing the song "Sincere," allowing Hill to slip away into the crowd. As they complete the last verse, the singers stroll into a wide shot with a fireworks display in the background. However, the skybursts are oddly magnified and erratically shift positions against the background, revealing that the fireworks display is a large but clumsily-edited rear-screen projection.
Answer: The movie doesn't take place in Indiana. It takes place in River City, Iowa, which is established multiple times throughout the movie.