President John F. Kennedy said on April 28, 1961 in Chicago, "If all of you had voted the other way - there's about 5500 of you here tonight - I would not be President of the United States."
Theodore White, in his book The Making of the President 1960 wrote, "this margin of popular vote is so thin as to be, in all reality, nonexistent. If only 4500 voters in Illinois and 28000 voters in Texas changed their minds, the sum of their 32000 votes would have moved both these states, with their combined fifty-one electoral votes, into the Nixon column...The election of 1960 can, if one wills, be seen as an interlocking set of ifs..." See pages 350-365 for further information.
On page 628 of his book JFK the Man & the Myth , Victor Lasky cites a 1961 GOP report that states "a record-breaking 69 million voters went to the polls last year. Yet fewer than 12,000 persons actually determined which man won the Presidency. Only 24,000 more votes for Vice President Nixon - properly distributed, of course, among the five states of Hawaii, Illinois, Missouri, New Mexico and Nevada - would have given him the needed electoral margin. Thus a switch of only 11871 votes cost Richard Nixon the White House."
Thomas Reeves in his book A Question of Character states (on pages 214-215) that "a mere 112,803 votes separated the two candidates - the smallest margin of the century. If only 4500 voters in Illinois and 24,000 voters in Texas had changed their minds, Nixon would have been president. In eleven states, a shift of less than 1 percent of the vote would have switched the state's electoral votes."