In highly resistive superconducting tunnel junctions, excess subgap current is usually observed and is often attributed to microscopic pinholes in the tunnel barrier. We have studied the subgap current in superconductor-insulator-superconductor (SIS) and superconductor-insulator-normal-metal ( SIN) junctions. In Al/AlOx/Al junctions, we observed a decrease of 2 orders of magnitude in the current upon the transition from the SIS to the SIN regime, where it then matched theory. In Al/AlOx/Cu junctions, we also observed generic features of coherent diffusive Andreev transport in a junction with a homogenous barrier. We use the quasiclassical Keldysh-Green function theory to quantify single- and two-particle tunneling and find good agreement with experiment over 2 orders of magnitude in transparency. We argue that our observations rule out pinholes as the origin of the excess current.