The full box space of a finitely generated group \(G\) is defined as the metrized disjoint union of all finite quotients of \(G\); the metric in each finite quotient \(G/N_i\) is the word metric coming from a fixed finite generating set of \(G\), and the sum of the diameters of \(G/N_i\) and \(G/N_j\) for elements in different finite quotients. “It is an interesting pursuit to find relations between the properties of the group and properties of the box space. These relations can be used to create metric spaces with strange properties in coarse geometry. Therefore, it is quite useful to study these space up to coarse equivalence.” In the present paper, the author considers free and free abelian groups. His results imply that the full box space of a free group \(F_k\) of rank \(k\) is not coarsely equivalent to the full box space of a free group \(F_d\) of rank \(d \geq 8k+10\) by showing that one of the box spaces has too many small components with respect to the other. It remains open whether for every box space of \(F_d\) (i.e., with respect to some subcollection \(N_i\) of finite index subgroups) there exists some box space of \(F_k\) which is coarsely equivalent to it. A second main result of the paper states that, for \(n \geq 3\), the full box space of \(\mathbb Z^n\) is not coarsely equivalent to the full box space of any 2-generated group.