For perspective, the largest college marching band currently is the James Madison University Royal Marching Dukes with 485. There is not a breakdown by section, but they probably fall slightly short of a trombonal level of 76.

Large bands in parades tend to suffer acoustic difficulties due to the distances involved. The speed of sound is 343m/s and if the band is really spread out you get the problem that you hear the sounds from the far end of the band later than the near edge of the band. In a parade it is important to keep the drums near the middle, but it still becomes difficult to synchronize. In some circumstances the musicians are instructed to watch the feet of the drummers rather than listen to the beat.

In the Macy's parade, bands go as wide as 16 abreast. Rows can be spaced about every two meters. At a march tempo of 120 beats per minute you get to be a quarter beat off after 43 meters. If you want everyone within 43 meters of the drummers it gives you a maximum size of 688 playing cohesively in a parade.

There are ways of working around this in a seated band. They can sit in a circle, and they can be using visual cues, and in some cases multiple conductors to keep together.

The lyrics of the song go on to mention thousands of reeds and many other instruments,

He then goes on to mention the bassoons. This is pretty unlikely unusual for a parade situation.

While very large bands certainly did exist in that time period (led by men like Gilmore and Sousa), smaller regional bands were certainly more common. Of note are a series of national band contests. Although they didn't feature bands as big as those described in The Music Man, the scope of them certainly inspired "76 Trombones" and songs like it. One must always remember that in the musical, Harry Hill is a pretty shady salesman, and was really trying to drum up as much business as possible.

Sources:

Emil A. Holz, "The Schools Band Contest of America (1923)", Journal of Research in Music Education Vol. 10:1 (Spring) p. 3.

James E. Moore, "National School Band Contests Between 1926 and 1931", Journal of Band Research 22:2 (Summer 1972) p. 233-234.

In short yes they did. I do not have my sources with me at the moment but I will update you later with the proper information. For now if you google "World Peace Jubilee of 1872" you can find some information on Patrick Gilmore and some of his large bands. Hope that can hold you over.

In the 2nd half of the 19th century there was an American phenomenon known as “monster concerts.” At one in 1872, the World Peace Jubilee, Johan Strauss conducted his “Blue Danube Waltz” with 2,000 players and 20,000 singers. He described the occasion (note the wonderful detail that the downbeat was signaled by a cannon!):

On the musicians' tribune there were twenty thousand singers; in front of them the members of the orchestra-and these were the people I was to conduct. A hundred assistant conductors had been placed at my disposal to control these gigantic masses, but I was only able to recognize those nearest to me, and though we had rehearsals there was no possibility of giving an artistic performance, a proper production. . . . Now just conceive of my position face to face with a public of four hundred thousand Americans. There I stood at the raised desk, high above all the others. How would the business start, how would it end? Suddenly a cannon-shot rang out, a gentle hint for us twenty thousand to begin playing The Blue Danube. I gave the signal, my hundred assistant conductors followed me as quickly and as well as they could and then there broke out an unholy row such as I shall never forget. As we had begun more or less simultaneously, I concentrated my whole attention on seeing that we should finish together too! Thank Heaven, I managed even that. It was all that was humanly possible. The hundred thousand mouths in the audience roared applause and I breathed a sigh of relief when I found myself in fresh air again and felt the firm ground beneath my feet."

Source (with text at Google Books): Daniel Cavicchi, [Listening and Longing: Music Lovers in the Age of Barnum] (https://books.google.com/books?id=Q9KQVfW2JwMC&pg=PT136&lpg=PT136&dq=mass+playing+of+blue+danube+cannon&source=bl&ots=q57qWO8ojQ&sig=z8Li5kuL8vnPjZmirBZ6WX4oBC0&hl=en&sa=X&ei=7lxzVfGuA4KuogSGt4GgCg&ved=0CB8Q6AEwAA#v=onepage&q=mass%20playing%20of%20blue%20danube%20cannon&f=false)

EDIT: Fixed the numbers.