I tried to do it in 19 days with a friend, which is about 27 miles per day. We did 14 days and 350+ miles, and then bailed. I feel safe to say that 27 mpd is TOO many miles per day on the CT, at least for me (at least without a zero or near-o day or two scattered in among them). There are a fair number of ups and downs - it seems like just when you get up above timberline and get a good view, it's time to go back down, as if the trail is afraid to stay up in the thunderstorm danger zone or something!
A related note is that I was pretty darn disappointed with the CT. I think some 40% of it is on old roads, and thus it doesn't necessarily go where a trail will go (e.g. along the side of the slope, with great views) but instead often where a road will go (e.g. in the middle of the ridgeline, in the forest, with no views). In general, lots of road walking is not something I'm too into.
To me, it felt a lot like what it is... a cobbled together assemblage of old logging roads, trails, 14er access, and whatnot. Of course, we did quit before the best part, the San Juans. However, I would never recommend doing a thru-hike of this trail to someone, when I think many better trails and routes exist out there.
There are some 14ers very close to the trail, so doing a side hike up a few of them would definitely be possible. And definitely plan for a long soak at the hot springs.