When a two-headed molecular motor such as kinesin is attached to its track by just a single head in the presence of an applied load, thermally activated head detachment followed by rapid re-attachment at another binding site can cause the motor to 'hop' backwards. Such hopping, on its own, would produce a linear force-velocity relation. However, for kinesin, we must incorporate hopping into the motor's alternating-head scheme, where we expect it to be most important for the state prior to neck-linker docking. We show that hopping can account for the backward steps, run length and stalling of conventional kinesin. In particular, although hopping does not hydrolyse ATP, we find that the hopping rate obeys the same Michaelis-Menten relation as the ATP hydrolysis rate. Hopping can also account for the reduced processivity observed in kinesins with mutations in their tubulin-binding loop. Indeed, it may provide a general mechanism for the breakdown of perfect processivity in two-headed molecular motors.