It's hard to remember a time when Jon Stewart wasn't the host of The Daily Show, but for a few years in the program's original run, Craig Kilborn was the host of the show before leaving in 1998. Several comedians auditioned for the job to take over as host, including Stephen Colbert, who was a correspondent on the show at the time. That job eventually went to Jon Stewart, but Colbert apparently set up a sting operation to find out why he didn't get the job.
In the summer of '98, when we announced that Jon was going to take over The Daily Show, we had a little press conference in the lobby of the old Comedy Central offices. And Stephen Colbert showed up, as a member of the press representing The Daily Show, wanting to know why he didn't get the job.
Collected from the upcoming The Daily Show (The Book) by Chris Smith (via an adaptation in a fascinating oral history by Vanity Fair), Doug Herzog (the president of Comedy Central at the time) remembers when they first announced that Jon Stewart was taking over the show. They announced the host change in 1998 and among the members of the press in attendance was Stephen Colbert. Colbert was at the Daily Show press conference as a representative for The Daily Show, and asked why he was passed over for the job.
Jon Stewart also apparently told Stephen Colbert he simply wasn't funny enough. This was more than likely done in jest, and I doubt there was any real animosity between the two men. Everything would go on to work out pretty well for the both of them (not that they knew that at the time). Stewart turned The Daily Show into a political lightning rod and gave it its biting voice, while Colbert went on to host his own widely successful spin-off in The Colbert Report before moving to host The Late Show on CBS.
If you've got the time then you should definitely give the Daily Show Oral History a read. It covers much of the uneasiness of Jon Stewart's initial takeover and the years leading up to when The Daily Show became an Emmy-winning juggernaut. It's got a collection of interviews from producers, correspondents, and guests from over the years, who reveal plenty of interesting behind-the-scenes details. There's also plenty of interviews with Stewart himself, who offers insight regarding changing the show to what he wanted, in opposition of the original writers who thought he was going to MTV it up.
You can catch The Daily Show with Trevor Noah when it airs on weeknights at 11 p.m. ET on Comedy Central.