"Texas and Illinois, the two largest states under dispute, witnessed the nastiest fights. In Texas, where Kennedy won the 24 electoral votes by a margin of 46,000 ballots,
the GOP took to the courts. But its suits were thrown out by a federal judge who claimed he had no jurisdiction. In Illinois, the appeal was pursued more vigorously, maybe because the electoral take was higher (27) and Kennedy's margin slimmer (9,000 votes).
Charges focused on Cook County (specifically Chicago) where Kennedy had won by a suspiciously overwhelming 450,000 votes.
National GOP officials plunged in. Thruston Morton flew to Chicago to confer with Illinois Republican leaders on strategy, while party Treasurer Meade Alcorn announced Nixon would win the state. With Nixon distancing himself from the effort, the Cook County state's attorney, Benjamin Adamowski, stepped forward to lead the challenge. A Daley antagonist and potential rival for the mayoralty, Adamowski had lost his job to a Democrat by 25,000 votes. The closeness of his defeat entitled him to a recount, which began Nov. 29.
Completed Dec. 9, the recount of 863 precincts showed that the original tally had undercounted Nixon's (and Adamowski's) votes, but only by 943, far from the 4,500 needed to alter the results. In fact, in 40 percent of the rechecked precincts, Nixon's vote was
overcounted. Displeased, the Republicans took the case to federal court, only to have a judge dismiss the suits. Still undeterred, they turned to the State Board of Elections, which was composed of four Republicans, including the governor, and one Democrat. Yet the state board, too, unanimously rejected the petition, citing the GOP's failure to provide even a single affidavit on its behalf. The national party finally backed off after Dec. 19, when the nation's Electoral College certified Kennedy as the new president—but
even then local Republicans wouldn't accept the Illinois results...
The GOP's failure to prove fraud doesn't mean, of course, that the election was clean. That question remains unsolved and unsolvable. But what's typically left out of the legend is that multiple election boards saw no reason to overturn the results. Neither did state or federal judges. Neither did an Illinois special prosecutor in 1961. And neither have academic inquiries into the Illinois case (both a 1961 study by three University of Chicago professors and
more recent research by political scientist Edmund Kallina concluded that whatever fraud existed wasn't substantial enough to alter the election)..."
http://www.slate.com/articles/news_and_politics/history_lesson/2000/10/was_nixon_robbed.2.html
