It can happen at any age, but it only happens drastically to players in their first three seasons. You will only see the 4 and 5 point jumps in speed to young guys (guys in their 1st 3 years). After that it'll only be 1 point increases, and those are sparse.
The problem is if you have a player that is between 90-92 speed in his 1st year, it is very likely that he will increase 3-4 points in speed each of the next two seasons. (Aka 97-99 speed rating)
My problem with this is that guys can get faster, but you'll never see a guy completely change his speed category. You will never refer to a power back with decent speed out of college as the fastest player in the NFL after a couple of seasons. (Jonathan Stewart in the 2nd year of my franchise)