  We generalize Y. Shi and L.-F.\ Tam's \cite{ShiTam} nonnegativity result for
the Brown-York mass, by considering nonnegative scalar curvature (NNSC)
fill-ins that need only be complete rather than compact. Moreover, the NNSC
fill-ins need not even be complete as long the incompleteness is ``shielded''
by a region with positive scalar curvature and occurs occurs sufficiently far
away.
  We accomplish this by generalizing P.~Miao's~\cite{Miao02} positive mass
theorem with corners to asymptotically flat manifolds that may have other
complete ends, or possibly incomplete ends that are appropriately shielded. We
can similarly extend other results on the compact NNSC fill-in problem to allow
for complete (or shielded) NNSC fill-ins. In particular, we prove the following
generalization of a theorem of Miao~\cite{Miao20}: Given any metric $\gamma$ on
a closed manifold $\Sigma^{n-1}$, there exists a constant $\lambda$ such that
for any complete (or shielded) NNSC fill-in $(\Omega^n, g)$ of
$(\Sigma^{n-1},\gamma)$, we have $\min_\Sigma H \le \lambda$, where $H$ is the
mean curvature of $\Sigma$ with respect to $g$.
