Sidney Bechet made a pair of famous overdubbed sides in 1941, "The Sheik of Araby" and "Blues of Bechet". Multi-instrumentalist Bechet recorded on six different instruments; each version had to be recorded onto a new master disc along with the preceding performance, with consequent loss of audio quality. The novelty was issued as "Sidney Bechet's One Man Band". The American Federation of Musicians protested the recording, putting an end to experiments with commercial overdubbing in the United States for years
Source: http://en.wikipedia.org/wiki/Overdubbing
On April 18, 1941, as an early experiment in overdubbing at Victor, Bechet recorded a version of the pop song "The Sheik of Araby", playing six different instruments: clarinet, soprano saxophone, tenor saxophone, piano, bass, and drums. A hitherto unissued master of this recording was included in the 1965 LP Bechet of New Orleans, issued by RCA Victor as LPV-510. On the liner notes, George Hoeffer quotes Bechet as follows:
I started by playing The Sheik on piano, and played the drums while listening to the piano. I meant to play all the rhythm instruments, but got all mixed up and grabbed my soprano, then the bass, then the tenor saxophone, and finally finished up with the clarinet.Source: http://en.wikipedia.org/wiki/Sidney_Bechet