Here's the plot of the movie from the Wikipedia entry:
The plot revolves around young Bart Collins [!?!?--CF: check out the author of the Well Tempered Blog for a rather significant coincidence], who lives with his widowed mother Heloise. The major blight on Bart's existence is the hated piano lessons he is forced to endure under the tutelage of the autocratic Dr. Terwilliker. Bart feels that his mother has fallen under Terwilliker's sinister influence, and gripes to visiting plumber August Zabladowski, without much result. While grimly hammering away at his lessons, Bart dozes off and enters a fantastical musical dream, in much the same fashion as Judy Garland in The Wizard of Oz.
In the dream, Bart is trapped at the surreal Terwilliker Institute, where the piano teacher is now a madman dictator who has locked up all non-piano-playing musicians in a dungeon and constructed a piano so large that it requires Bart and 499 other enslaved boys (the aforementioned 5,000 fingers) in order to play it. Bart's mother has been turned into Terwilliker's hypnotized assistant and bride-to-be, and Bart must dodge the Institute's guards as he scrambles to save both his mother and himself. He tries to recruit Mr. Zabladowski, who has been hired to install all of the Institute's sinks ahead of a vital inspection, but only after much skepticism and foot-dragging is the plumber finally convinced to help. The two of them empty their pockets and construct a noise-sucking contraption which ruins the mega-piano's opening concert. The enslaved boys cheerfully run riot, and the "VERY atomic" noise-sucker explodes in spectacular fashion, bringing Bart out of his dream.
The movie ends on a hopeful note for Bart, when the real-life Mr. Zabladowski finally notices Heloise, and offers to drive her into town in his jeep. Bart escapes from the piano, and triumphantly runs off to play.
(Via Dr. B)