People who worked with Jon Stewart on The Daily Show tell what it was like.
You'll notice it mentions that at one point, Stewart was signed by David Letterman's Worldwide Pants company. This was when Tom Snyder was hosting the program following Dave and the industry buzz at the time was that signing Stewart was someone's way of telling Ol' Tom it was time to retire again. The assumption was that Dave (or someone) wanted Stewart there to "young up" the demographic and maybe to be in the "on deck" circle if and when Dave decided to retire.
If it was intended as a hint at the time, Snyder didn't take it, not even after Stewart filled in for him a few times. When Snyder was finally nudged aside, it was not Stewart that got the job but Craig Kilborn. He left The Daily Show to do it and guess who they got to replace him over there.
I'd be curious to know if Stewart thought, as many did, that Worldwide Pants signed him with no intention of ever giving him the show after Letterman's, let alone Letterman's. A lot of people in the business seem to think the whole idea of signing Stewart was to keep him off the air lest he compete with Dave either by going on opposite him or following him and looking like a viable replacement. I suspect Mr. Stewart is too much the gentleman to ever say that's what he thought was done to him but I wonder if he thinks that.