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Cobbs was born in Smale, Arkansas and moved to Chicago, Illinois, in 1951, occasionally performing in local clubs with Little Walter, Eddie Boyd and others. He served in the American armed forces and then returned to Chicago, recording a number of singles on such labels as Ruler, a subsidiary of J.O.B. Records.
He first recorded his own composition "You Don't Love Me" in 1960 for Mojo Records, a record label in Memphis, Tennessee owned by Billy Lee Riley. It was then leased to Vee-Jay Records for release. The song has since been covered by various artists including The Allman Brothers Band, Grateful Dead, Richie Kotzen, Kaleidoscope, Quicksilver Messenger Service, John Mayall & the Bluesbreakers, Junior Wells and Magic Sam, as well as by Mike Bloomfield, Al Kooper and Stephen Stills on their 1968 Super Session. Another cover of "You Don't Love Me" was the 1967 rocksteady rendition by Dawn Penn known as "You Don't Love Me (No, No, No)".
Cobbs later released singles on a wide variety of labels, and ran nightclubs in Arkansas and Mississippi through the 1970s and 1980s. He went on to release the albums Hey Little Girl for the Wilco label in 1986, and Down to Earth on the Rooster Blues label in 1994. Cobbs has performed at the King Biscuit Blues Festival and the Chicago Blues Festival. Cobbs also appeared in the 1991 film Mississippi Masala; he performed the songs "Angel from Heaven" and "Sad Feelin'" for the film.