If I'm not mistaken, the [Reeb stability theorem][1] prevents this from happening for $n\geqslant 3$. There is the following version of it, contained in: C. Godbillon. _Feuilletages, etudies geometriques_, Théorème 3.1:

> **Theorem (Reeb global stability).** Let $\mathcal F$ be a codimension $1$ foliation of a compact connected manifold; if the boundary of $M$ is non-empty, let $\mathcal F$ be transverse or tangent to the boundary. If $\mathcal F$ has a compact leaf with finite fundamental group, then all the leaves are compact and have finite fundamental group.

and there is a local version saying that such a compact leaf with finite fundamental group is stable. These together imply that the leaves of $\mathcal F$ between two spheres $S_1$ and $S_2$ are all spheres (the global version ensures compactness, and the local version says that locally they are all covering spaces of spheres and therefore spheres).

I guess, the key difference from the case of circles is that the circle has infinite fundamental group, so one cannot use Reeb stability in that case.

  [1]: https://en.wikipedia.org/wiki/Reeb_stability_theorem