both academic and athletic scholarships are considered "institutional aid."
there is an NCAA threshold for "academic scholarship". bylaw 15.5.3.2.1 you have to be either:
-top 10% of class
-3.50 GPA
or 1200 SAT.
If they receive 1 penny of athletic aid, and they also receive institutional "academic money" and they don't meet one of those above criteria, then all that "academic money" is called "countable aid" and acts just like an athletic grant.
SO, in your hypothetical, if the kid meets any of the above criteria, then the coach only has to worry about the value of the "athletic scholarship" they want to give. But if the kid doesn't meet one of those criteria, then the coach has to watch both numbers in total.
Plenty of kids who are on 50 or 60% athletic aid with great grades that have total aid approaching 100%.
The problem is when a school gives academic money to kids with 3.3 and 1100 SAT. Those are nice marks, but all that academic money is going to count against the coach if he adds on ANY athletic money. So it just becomes a matter of how much the coach values the kid for the team. Sometimes you have to tell a kid, take the academic money because it is more than I can give you athletically, but in your case, i can't "stack the money."
Your hypothetical has some complications to it that don't allow for one single answer. It is case by case, based on many variables. You CAN end up with a "full" using both means. But it is often a hard concept for parents to grasp that sometimes they can't have both.