The wave trace contains information on the asymptotic distribution of eigenvalues for the Laplacian on a Riemannian manifold. It is well known that its singular support is contained in the length spectrum, which allows one to infer geometric information only under a length spectral simplicity or other nonresonance type condition. We construct large families of domains for which there are multiple geodesics of a given length having different Maslov indices, leading to a cancellation of arbitrarily many orders in the wave trace at that length. This shows that there are potential limitations in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different objects, at least insofar as the wave trace is concerned.