Fix $n \geq 3$ and let $S \subset \mathbb{R}^n$ be a smoothly embedded $2$-sphere. Must there exist a smoothly embedded $3$-ball $B \subset \mathbb{R}^n$ such that $\partial B = S$? This is true for $n=3$ by the Schoenflies theorem. Also, it is true for $n \geq 7$; indeed, since $\mathbb{R}^n$ is contractible we can find a continuous map $B \rightarrow \mathbb{R}^7$ taking $\partial B$ homeomorphically onto $S$, and by Whitney's theorem we can jiggle this continuous map a little to make it an embedding. This might also work for $n=6$, but the homotopy to make $B$ embedded might be complicated and mess up $S$ (I don't know the details of Whitney's strong embedding theorem), so I'm not sure.

What about $4 \leq n \leq 6$?