The answer to this question for any $n\geq 4$ is positive iff it is positive for $n=4$. The question is similar to the question of existence of a polyhedron $N$, which is a topological $n$-manifolds but is not a PL $n$-manifold. By considering a link $L_a(N)$ at a point $a\in N$, one would need $L_a(N)$ to be a topological $(n-1)$-sphere not PL-equivalent to the usual $(n-1)$-sphere. For $n\geq 6$, any PL $(n-1)$-manifold homeomorphic to $S^{n-1}$ is PL homeomorphic to $S^{n-1}$. Thus, for $n\geq 6$, the only possibility for $L_a(N)$ not to be PL homeomorphic to $S^{n-1}$ is not to be a PL $(n-1)$-manifold. That's how the problem is reduced to the dimension one less. Finally, if there exists $X^4$ a smooth exotic $S^4$, then by considering its induced PL structure (recall that in dimension 4 any PL manifold admits a unique compatible smooth structure), we get that its $(n-4)$-iterated suspension $\Sigma^{n-4}X^4$ is a polyhedron homeomorphic to $S^n$, but is not a PL $n$-manifold, $n\geq 5$.

The same thing happens for the Schoenflies problem. If there is a Schoenfiles PL embedding $f\colon S^3\hookrightarrow S^4$, which is not PL equivalent to the equatorial inclusion, then $\Sigma^{n-4}f\colon S^{n-1}\hookrightarrow S^n$ is not PL locally flat, $n\geq 5$, by induction. (Note that it would still be topologically locally flat, since $f$ is trivial topologically.) Otherwise, any such embedding is PL locally flat by induction, and therefore is trivial, see page 47 in "Introduction to piecewise-linear topology" by Rourke-Sanderson.