  In this article, we study the relationship between the weak limit of a
sequence of integral currents in a metric space and the possible Hausdorff
limit of the sequence of supports. Due to cancellation, the weak limit is in
general supported in a strict subset of the Hausdorff limit. We exhibit
sufficient conditions in terms of topology of the supports which ensure that no
cancellation occurs and that the support of the weak limit agrees with the
Hausdorff limit of the supports. We use our results to prove countable
$\hm^m$-rectifiability of the Gromov-Hausdorff limit of sequences of Lipschitz
manifolds $M_n$ all of which are $\lambda$-linearly locally contractible up to
some scale $r_0$. In an appendix, written by Raanan Schul and the second
author, it is shown that the Gromov-Hausdorff limit need not be countably
$\hm^m$-rectifiable if the $M_n$ have a common local geometric contractibility
function which is only concave (and not linear). We also relate our results to
work of Cheeger-Colding on the limits of noncollapsing sequences of manifolds
with nonnegative Ricci curvature.
