If a quarterback has a chance to be a star, chances are it will become apparent within two seasons. Chase Stuart of Football Perspective looked at how long it took star quarterbacks to become great, examining the top 42 quarterbacks since the merger and compared how each quarterback performed (measured by adjusted net yards per attempt) relative to the league average during their career. He then broke down their early careers into segments of 16 stars to reflect the current NFL season duration.
Of the 42 quarterbacks studied, only four had below-average passing stats after two years of starting. Those four players were Terry Bradshaw, Troy Aikman, Vinny Testaverde, and Drew Brees.
As Stuart puts it:
If a quarterback is below-average through two years worth of starts — say, Ryan Tannehill— then it seems highly unlikely that such a player will turn into a franchise quarterback absent extenuating circumstances.
The "extenuating circumstances" gives for Bradshaw, Aikman, Testaverde, and Brees were that each landed with terrible teams.
Note that this isn't telling us who will be a great quarterback, as Stuart didn't track down all the QBs who are at least above average through their first two years. This is sussing out a negative indicator, showing which QBs have virtually no chance of getting to the highest level of quarterback.
This raises further questions—how does his manifest at different cutoffs, both in starts and for caliber of quarterback?—but it's a compelling case that there's a set timer on a quarterback to get himself and his offense to an above average level if he's going to be great.