Jeopardy! has a very distinct format, and it's very rare the long-running game show breaks from its traditional camera shots. Such consistency and tradition creates some unique problems for the show, and means that, apparently, a contestant can potentially be turned away from the show for being too tall.
The allegation is now out following a recent episode of Ken Jennings and John Roderick's podcast Omnibus. During an episode on the Patagonian Giants, Jennings mentioned his friend who he auditioned with to be on Jeopardy!. Jennings talked about the incident, his 6'8" friend's suspicion, and the reason why size matters during a taping of the game show.
We both passed the test. I think he feels he was too tall for the show. Kareem [Abdul-Jabbar] has been on Jeopardy!, but typically they try to make everybody be the same height. They want the pan across the three contestants at the end not to be [sloping noise].
Ken Jennings pointed out that while some tall people (Kareem Abdul-Jabbar is 7'2") have appeared on Jeopardy!, the idea is that all three contestants be around the same height for logistical purposes. Otherwise, the pan shot at the game's end can't be a straight line, which would create a sloping shot that could look sloppy to audiences compared to the former option. For that reason, Jennings stated that there's an effort to make every contestant as close to the same height as possible.
Jeopardy! used to have a way to accommodate taller contestants, but it was far from ideal. Ken Jennings explained that there were trap doors to help the taller contestants appear shorter, but that ended up being more of a hazard than help.
Usually they put the others [shorter people] on boxes. Originally they had trap doors where they could lower people, but people kept falling down. You don’t want to play a game show in a hole. Now, the nice short lady can fall off her box. It’s a tough situation.
Tall people tripping and falling in holes is bad, but so is short people falling down from boxes while attempting a handshake. It's kind of amazing how much of a problem this can be for Jeopardy!, and yet, most audiences at home would have no idea there was a problem while watching Alex Trebek give out the questions. By the way, for anyone curious, Jennings is around 5'10, and James Holzhauer is 5'8. The average height in America is 5’9 for men and 5’4 for women, so perhaps this is the sweet spot for a Jeopardy! contestant? The 6’0 frame of Brad Rutter (who still has a big Jeopardy! record) may dispute that, but we all saw how he did in the GOAT tournament.