We formulate a conjecture for the almost rigidity statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. Furthermore we establish the conjecture under the assumption of spherical symmetry in all dimensions. In particular, it is shown that a sequence of asymptotically flat initial data satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the sense that the bases converge in the volume preserving intrinsic flat sense to Euclidean space. Furthermore, the difference of the second fundamental forms must converge to zero in . This is joint work with E. Bryden and C. Sormani.