Disordered packings of unbonded, semiflexible fibers represent a class of materials spanning contexts and scales. From twig-based bird nests to unwoven textiles, bulk mechanics of disparate systems emerge from the bending of constituent slender elements about impermanent contacts. In experimental and computational packings of wooden sticks, we identify prominent features of their response to cyclic oedometric compression: nonlinear stiffness, transient plasticity, and eventually repeatable velocity-independent hysteresis. We trace these features to their micromechanic origins, identified in characteristic appearance, disappearance, and displacement of internal contacts.