Yes, kids cognitive abilities improve the longer they have been in school, but it's certainly plausible that better cognitive abilities make it more probable that you'll stay in school longer. And schooling is also confounded with age--kids who have been in school longer are also older and therefore have had more life experiences, and perhaps those have prompted the increases in intelligence.
One strategy is to test everyone on their birthday. That way, everyone should have had the same opportunity for life experiences, but the student with a birthday in May has had four months more schooling than the child with the January birthday.
That solves some problems, but it entails other assumptions. For example, older children within a grade might experience fewer social problems, for example.
The authors capitalized on the fact that every male in Sweden must take a battery of cognitive tests for military service. The testing occurs near his 18th birthday, but the precise date is assigned more or less randomly (constrained by logistical factors for the military testers). So the authors could statistically control for the time-of-year effect of the birthday and in addition investigate the effects of just a few days more (or less) of schooling. The researchers were able to access a database of all the males tested between 1980 and 1994.
The authors found that older students scored better on all four tests--no surprise there. What about students who were the same age, but who, because of the vagaries of the testing, happened to have had a few days more or fewer of schooling?
More schooling was associated with better performance, but only on the crystallized intelligence tests: an extra 10 days in school improved by about 1% of a standard deviation. Extra non-school days had no effect.
There was no measurable effect of school days on the fluid intelligence tests. This result might mean that these cognitive skills are unaffected by schooling, but it might also mean that the "dose" of schooling was too small to have an impact, or that the measure was insensitive to the effect that schooling has on fluid intelligence.
Carlsson, M. Dahl, G. B. & Rooth, D-O. (2012). The Effect of Schooling on Cognitive Skills. NBER Working Paper No. 18484 October 2012