We analyze the Casimir force between two parallel infinite metal cylinders, with nearby metal plates (sidewalls), using complementary methods for mutual confirmation. The attractive force between cylinders is shown t ohave a nonmonotonic dependence on the separation to the plates. Thisintrinsicallymulti-bodyphenomenon, which occurs with either one or two sidewalls(generalizing an earlier result for squares between two sidewalls), does not follow from any simple two-body force description. We can, however, explain the nonmonotonicity by considering the screening (enhancement) of the interactions by the fluctuating charges (currents) on the two cylinders, and their images on the nearby plate(s). Furthermore, we show that this effect also implies a nonmonotonic dependence of the cylinder-plate force on the cylinder-cylinder separation.