We study infinite families of black hole microstates consisting of wormholes and shells of matter. They are orthogonal at leading order in the saddle point approximation of the Euclidean gravitational path integral, suggesting a dramatic overcounting of the dimension of the microcanonical subspace. However, wormhole contributions in higher moments of the overlaps reveal small off-diagonal components. This non-orthogonality reduces the (naively infinite) dimension of the space spaned by these states to the expected result: the exponential of the Bekenstein-Hawking entropy.