The below recursion tree will help you to get an idea of how many levels will suffice
Assuming fib(n) is denoted as F(n) in the diagram,
I have taken an example of F(4).
Now, you know function calling sequence in recursion is in preorder and evaluation order is postorder.
Each recursive call of F(i) takes 4 bytes of storage on the stack where 2 bytes are taken by parameter i and 2 bytes for the return address.
Even for the initial call, return address would be saved for F(i).
So, as we see from recursion tree, longest chain of recursion will be depicted by longest chain or leg of recursion tree and the number of levels in the recursion tree will tell, how many recursive calls will be active at a time in the stack.
Also, note, at each level, all the number of function calls will take only one unit of stack,
because function evaluation order being postorder,
LEFT, RIGHT, ROOT, return
So, now we need to find how many levels can we have in total where each level takes 4 bytes and total we can allow for 64 bytes of storage.
64/4=16 levels of recursion can be accommodated all at a time in the stack.