We consider geometrically infinite Kleinian groups and, in particular, groups with singly cusped parabolic fixed points. In order to distinguish between different geometric characteristics of such groups, we introduce the notion of horospherical tameness. We show that Kleinian groups with singly cusped parabolic fixed points are of 2-convergence type, and give a brief discussion of the fractal nature of their limit sets. Subsequently, we use Joergensen's analysis of punctured torus groups to relate horospherical tameness to diophantine properties of Thurston's end invariants.