I just looked at an electoral map of the 1960 election. You can see it here:
http://en.wikipedia.org/wiki/United_States_presidential_election,_1960

As you can see if you look at the page, Kennedy won the electoral vote by an 84 vote margin 303-219. At the time, Illinois had 27 electoral votes and West Virginia had eight, for a total of 35. If my math is correct, even if all the stories about cheating in those states is true, that means Nixon still would have lost by 49 electoral votes.