Twenty-seven is an eerie age for musicians. There are so many rock icons who died right before their 28th birthdays—even an abbreviated list includes Jimi Hendrix, Janis Joplin, Kurt Cobain, Amy Winehouse and Jim Morrison—there’s even a name for them. They are The 27 Club.

It turns out 27 is a special age in the NBA, too, except in the opposite way. This is the age when basketball players break through.

It...