We consider a directed polymer pinned by one-dimensional quenched randomness, modeled by the space-time trajectories of an underlying Markov chain which encounters a random potential of form u + V_i when it visits a particular site, denoted 0, at time i. The polymer depins from the potential when u goes below a critical value. We consider in particular the case in which the excursion length (from 0) of the underlying Markov chain has power law tails. We show that for certain tail exponents, for small inverse temperature \beta there is a constant D(\beta), approaching 0 with \beta, such that if the increment of u above the annealed critical point is a large multiple of D(\beta) then the quenched and annealed systems have very similar free energies, and are both pinned, but if the increment is a small multiple of D(\beta), the annealed system is pinned while the quenched is not. In other words, the breakdown of the ability of the quenched system to mimic the annealed occurs entirely at order D(\beta) above the annealed critical point.