The people who are blithely saying "range doesn't improve" should take a look at the UZR numbers from Drew when he was in his early 20s and see how shitty his defense was. The very guy that you're comparing Bogaerts to had the same exact issue when he first came up to the majors. After a few years Drew figured it out and played good defense when he was healthy (he had a few more bad years around his ankle injury).
Drew's UZR/150 by year (age):
-12.6 in 480.1 innings (age 23)
-13.9 in 1283.1 innings (age 24)
-17.7 in 1294.1 innings (age 25)
3.0 in 1142 innings (age 26)
10.1 in 1259.1 innings (age 27)
As noted by Savin Hillbilly, Pedroia's history tells a similar story (though he didn't start out nearly as poorly). There is more to defense and even to range than the athleticism of youth.
Drew's UZR/150 by year (age):
-12.6 in 480.1 innings (age 23)
-13.9 in 1283.1 innings (age 24)
-17.7 in 1294.1 innings (age 25)
3.0 in 1142 innings (age 26)
10.1 in 1259.1 innings (age 27)
As noted by Savin Hillbilly, Pedroia's history tells a similar story (though he didn't start out nearly as poorly). There is more to defense and even to range than the athleticism of youth.