I was thinking that it might involve some player setting a record and then extending it later in the same game. Then again, he hasn't been deposed as record holder, rather just the number had changed. Gretzky kept breaking his own record the year he scored 80, 81, 82 ... 92 goals, and that's not very interesting from this perspective.
You'd have to find a situation wherein two players were chasing a record and both managed to break it while they were still close.
Like, say, Mark McGwire and Sammy Sosa in baseball, 1998. The National League record for home runs was 56, and both of them passed that mark rather easily. I think that Sosa made it to 60 a day or so before McGwire did, though the latter would go to 70 on the season. So, Sammy held that record just some hours, and they swapped the Major League record of 61 once or twice. (Unless a seasonal record doesn't count until the season is complete, which is a reasonable argument).
I recall that Bernie Geoffrion and Frank Mahovlich challenged Maurice Richard's record of 50 goals the same season, in 1960-61. However, although Geoff got to 50, Frank came up short, so Mahovlich never held the record.
You'd have to find some situation like that.
I'd guess that something like penalty minutes per game might have been set multiple times on the same night, when there was a lot of fighting.