It's funny that 1960 is brought up as an example because it, along with 1976, are the two post-World War II elections where, IMO, the candidates were the most similar to each other. Both also happened to be decided by fairly close margins. The closest we have to that now would be the 2000 election, a near 50-50 split with similar dynamics to both 1960 and 1976.
Gore: 52
Bush: 65
Competitive: 421
So using Antonio V's formula of counting states that differ from the national PV margin by more than 10 points for either party, the most recent election with a similar result to 1960 was 2000, with more competitive electoral votes. The difference between the two candidates in PV was also almost exactly the same, as well as having a two-term VP heading the ticket for each incumbent party losing despite the popularity of the incumbent President. Some states are just "swingier" than others are, as 2008 shows.