  The (relativistic) center of mass of an asymptotically flat Riemannian
manifold is often defined by certain surface integral expressions evaluated
along a foliation of the manifold near infinity, e. g. by Arnowitt, Deser, and
Misner (ADM). There are also what we call 'abstract' definitions of the center
of mass in terms of a foliation near infinity itself, going back to the
constant mean curvature (CMC-) foliation studied by Huisken and Yau; these give
rise to surface integral expressions when equipped with suitable systems of
coordinates. We discuss subtle asymptotic convergence issues regarding the ADM-
and the coordinate expressions related to the CMC-center of mass. In
particular, we give explicit examples demonstrating that both can diverge -- in
a setting where Einstein's equation is satisfied. We also give explicit
examples of the same asymptotic order of decay with prescribed mass and center
of mass. We illustrate both phenomena by providing analogous examples in
Newtonian gravity. Our examples conflict with some results in the literature.
