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1919, American English, originally a reference to rumors of quadriplegics as a result of catastrophic wounds suffered in World War I (the military vehemently denied there were any such in its hospitals), from basket (n.) + case (n.2). Probably literal, i.e., stuck in a basket, but basket had colloquial connotations of poverty (begging) and helplessness long before this. Figurative sense of "person emotionally unable to cope" is from 1921.
[1960s+; fr a 1919 term describing a person, usu a wounded soldier, without either arms or legs, who needed to be carried in a basket; use revived in 1939 by Dalton Trumbo's novel Johnny Got His Gun]