We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. We prove theorems demonstrating when the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat limits of sequences of such metric spaces agree. Then for sequences of Riemannian manifolds with boundary we only require nonnegative Ricci curvature, upper bounds on volume, non collapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary to obtain converging subsequences where both limits coincide,