That cut rule went into effect beginning in 2012; for many years prior to that, the U.S. Open cut rule included the common "and all golfers within 10 strokes of the lead" provision. That provision meant that if a golfer was outsite the Top 60 - say, in 68th place - but was within 10 strokes of the leader's score, he still made the cut. However, that 10-strokes rule was eliminated beginning in 2012.
So to repeat: The U.S. Open cut rule is Top 60 plus ties. The "plus ties" provision means that there are no playoffs for the final spots in the Top 60. Say 58 players are at +5 or better, and another seven players are at +6. All seven of those golfers at +6 make the cut because of the "plus ties" provision, so in this example 65 golfers make the cut.
The U.S. Open first used a 36-hole cut in 1904.