We can restrict your problem to the case of open manifolds. It turns out that "finite dimensional integral singular homology" (i.e., finitely generated in each degree) is almost the same thing as the manifold being the interior of a compact manifold with boundary. For example, if $M$ is a 1-connected and open manifold of dimension $>5$, then the Browder-Levine-Livesay theorem says that $M$ is the interior of a compact manifold with boundary
(where the boundary is also 1-connected) iff the homology of $M$ is finitely generated and 
$M$ is $1$-connected at infinity. 

This result was later generalized in Siebenmann's thesis to the non-simply connected case.

**Addendum.** Here's a link to Siebenmann's thesis: 

www.math.uchicago.edu/~shmuel/tom-readings/Siebenmann%20thesis.pdf