Some (topological) manifolds can be embedded as the interior of a compact manifold with boundary. Any closed manifold, for example, or any closed manifold with some points removed, and so on.
On the other hand, some cannot. For example, as far as I can tell, an orientable surface of infinite genus cannot. This has infinite-rank second homology group, which cannot happen for the interior of a compact manifold with boundary.
Perhaps this is a naive question, but is there an easy way to tell, perhaps using some invariants like homotopy groups or (co)homology, if a given topological manifold (without boundary) can be embedded as the interior of a compact manifold with boundary?
Given a $k$-manifold $M$, a necessary condition that it be the interior of a compact manifold with boundary is that there exists a compact submanifold with boundary $C \subset M$ such that the pair $(M-int(C),\partial C))$ is homeomorphic to the pair $(\partial C \times [0,\infty),\partial C \times 0)$.
Perhaps the most well known counterexample is the Whitehead manifold, a contractible 3-dimensional manifold.
In dimension $k \ge 5$, Siebenmann's thesis gives a complete characterization (there seem to be several links on Wikipedia to Siebenmann's thesis, not sure which is the best).