  We explore the distinctions between $L^p$ convergence of metric tensors on a
fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat
convergence of the corresponding sequence of metric spaces. We provide a number
of examples which demonstrate these notions of convergence do not agree even
for two dimensional warped product manifolds with warping functions converging
in the $L^p$ sense. We then prove a theorem which requires $L^p$ bounds from
above and $C^0$ bounds from below on the warping functions to obtain enough
control for all these limits to agree.
