Hyperbolic-type metrics extend the idea of negative curvature to metric spaces, and several well-behaved hyperbolic-type metrics are known on 1-punctured Euclidean space. However, they lose their hyperbolicity on spaces with non-singleton boundary. In this talk, I will discuss the obstructions to hyperbolicity on more general boundaries, and give a recent result which allows hyperbolicity over n-punctured Euclidean space.