A study is made of the constraints placed upon the couplings and resonance spectrum required to saturate finite-energy sum rules consistently in the region around t=0 . It is argued that the probable existence of crossing trajectories and cuts places a fundamental limit on the usefulness of finite-energy sum rules. The demand that the spin of the resonances increase monotonically with mass implies that an infinite number of asymptotically parallel daughter trajectories in the resonance region are required for consistency. This result is independent of any statement about the energy dependence of the leading trajectory.