Motivated by an analogy with traffic, we simulate two species of particles ("vehicles"), moving stochastically in opposite directions on a two-lane ring road. Each species prefers one lane over the other, controlled by a parameter 0 <= b <= 1 such that b = 0 corresponds to random lane choice and b = 1 to perfect "laning". We find that the system displays one large cluster ("jam") whose size increases with b, contrary to intuition. Even more remarkably, the lane "charge" (a measure for the number of particles in their preferred lane) exhibits a region of negative response: even though vehicles experience a stronger preference for the "right" lane, more of them find themselves in the "wrong" one! For b very close to 1, a sharp transition restores a homogeneous state. Various characteristics of the system are computed analytically, in good agreement with simulation data.