JFK doesn't choose LBJ as his running mate, instead choosing Stuart Symington creating a division in the ranks and potentially alienating Southern Democrats.
Especially plausible since his brother Bobby didn't get along with LBJ at all.
"Fear and Loathing in the White House"
http://www.nytimes.com/books/97/10/26/reviews/971026.26oshinkt.html
As Jeff Shesol notes in ''Mutual Contempt,'' a penetrating and richly detailed account of the ''feud'' that shaped the 60's, [Robert] Kennedy despised Johnson with a ferocity that startled many observers, while Johnson harbored fears of Kennedy that bordered on paranoia.
...
The Robert Kennedy-Lyndon Johnson feud took on a life of its own. With little thought (and for reasons that still are unclear), John Kennedy chose Johnson to be his running mate. When news of this selection enraged key Northern liberals, Robert Kennedy was dispatched to Johnson's hotel suite to persuade him to withdraw. He failed in his task, but did earn Johnson's enmity as a ''grandstanding little runt.''