We show experimentally for the first time that two mutually attracting flux ropes may bounce back instead of merging together, leading to a variety of dynamics not expected from a two-dimensional model. Attraction forces due to flux rope currents compete with repulsion from field line bending of in-plane and out-of-plane magnetic fields and elastic plasma compression. Bouncing dynamics occurs if the line-bending force due to an out-of-plane field dominates. Otherwise, the ropes merge. Further reduction in the field line-bending force results in violently erratic magnetic states.