5 Kids really answered this question the best that you could have - Aikman was only as good as the team around him, and he provided a great example as far as David Carr goes.
To give you a couple of other examples of recent QB's who had a decent shot:
Ryan Leaf - 2 TDs to 15 INT's his rookie year. Benched for a year, Returned the following season to lead his team to a 1-8 record under the helm.... then he managed to get a starting position on another team the following season! 4-17 record as a starter.... and I know a couple of teams that toyed with bringing him in AGAIN!
Alex Smith - 1 TD to 11 INTS his rookie year. Played a full 2nd year, doing alright, leading the team to 7-9. 3rd year was awful - lost his starting job, and was cut this year for the immortal JT Sulivan and Shaun Hill.
Akilli Smith - 3-14 as a starter, but managed to be considered for a starting job for 3 seasons. in 17 starts (22 total appearances), he threw 5 TD's.
Brodie Croyle - 0-6 (or 8 now?) as a starter.... constantly injured, yet still may be the Chiefs "future."
Matt Leinhardt - Not a bad record as a starter, but definitely doesn't put up the number Kurt Warner does with the same talent around him. The franchise hasn't given up on him, but it hasn't given him the starting job either...
Eli Manning - Started off 1-6 in his career, and as recent as last November, mistakes made fans call for him to be removed as the starter....
Peyton Manning - Threw 28 INT's as a rookie, leading his team to a 3-13 record. Threw 15, 15, 23, and 19 the next 4 seasons on his way to becoming the best QB of his generation.
Although I definitely get your point, and see what you're saying, sometimes it is best for teams to pull the rug out from a QB who they simply don't think can make the cut. I do agree that often times teams will do that too early with their young QB's. Ironically the reason for this may be because of Aikman - he was a game managing QB who won a lot of Super Bowls. Trent Dilfer, Brad Johnson, and a few others followed this same mold, and they were successful. If a team is built with a good D and rushing attack, rather than risk it all on a young QB, it makes much more sense to put the game in the hands of someone who will make all the smart plays.