Part of what's motivating this question is the idea of putting "coordinates" on a manifold. The connection is that: given a manifold, if it is homeomorphic to an $n$-ball with parts of its surface identified, then it would be homeomorphic to an $n$-cube with parts of its surface identified. This allows us to impose a coordinate system on the manifold.
So, if the answer to the question is "no", then it seems that there would be a (connected compact) manifold we cannot put coordinates on.
In 2 dimensions, the answer is "yes", which is clear by looking at the fundamental polygon of the manifold.
A more general question would be whether every connected $n$-manifold can be constructed by starting with an open $n$-ball, adding some points on the surface, and identifying some of those surface points.