No, not one of the many songs with that name.
This thread was inspired by my re-watching The Incredible Shrinking Man on Svengoolie over the weekend. The movie is based on Richard Matheson’s novel The Shrinking Man, but when they made the movie (which they began doing even before the novel was published) they changed the title to “The Incredible Shrinking Man”, as if the title didn’t have enough oomph without that word. Maybe they thought people would think the film was about a guy who was just very, very shy.
The novel continued to sell under its original title until 1983, when it appeared as "The Incredible Shrinking Man in Britain, and again in 1988. It didn’t get the adjective in the US until the 1995 Tor edition. Obviously, the publishers were taking advantage of the more familiar movie title, even at that late date.
As far as I ca tell, only one movie prior to this had “Incredible” in the title – the 1942 short The Incredible Stranger. But introducing that word into the title of the science fiction movie based on Matheson’s book started a trend.
First, to compete with the success of the Universal B-movie, American International made a film about a man who grew, instead of shrinking. The Amazing Colossal Man came out. They used a different adjective, but they were clearly copying the title and the premise (both size changes are somehow the result of radioactivity, the usual 1950s bugaboo). They hammered home the idea in the following year’s sequel , War of the Colossal Beast (No “Amazing” this time, though. Even though the original Colossal Man was a low-budget quickie, this was even lower budget, as evidenced by the fact that they couldn’t persuade the original leads to star in it.)
But now we got a whole stream of “Incredible” movies, as if the titles would not be compelling enough to lure moviegoers in from their warm living rooms and TVs:
The Incredible Petrified World (1959)
The Incredible Journey (1963)
The Incredible Mr. Limpet (1964)
The Incredibly Strange Creatures who Stopped Living and became Mixed-up Zombies (1964)
The Incredible Two-Headed Transplant (1971)
The Incredible Invasion (1971)
The Incredible Sarah (1976)
The Incredible Hulk (1977)
The Incredible Melting Man (1977)
The Incredible Shrinking Woman (1981)
… and plenty of others in later years. Actually, The Incredible Hulk took its title from the Marvel comic, which started with that title in 1962.
But the trend of superfluous adjectives in titles continued with “Amazing” and “Fantastic” and others.
I have to admit, there were lots of titles with “Amazing” or “Fantastic” in them pre-1957. But it’s hard not to see the influence of “The Incredible Shrinking Man” in “The Amazing Transparent Man” (1960) or “The Fantastic Invasion of Planet Earth” (as the 1976 re-release of Arch Oboler’s 1966 3-D science fiction film The Bubble was called. )