Filamentary objects such as nano-wires, nanotubes and DNA are of current interest in physics, nanoscience, chemistry, biology and medicine. They can interact via strong, exceptionally long-ranged many-object van der Waals (vdW, dispersion) forces, causing them to cluster into multi-object bundles. We analyse their vdW interactions perturbatively, predicting N-object vdW energy contributions that alternate in sign with increasing N. Our findings are confirmed here via the first detailed analysis of a 4-cylinder vdW model. We also provide novel insights permitting these tendencies to be understood simply in terms of electronic screening and anti-screening. Our results suggest that a non-perturbative calculation will be required for reliable prediction of dispersion interactions in these ubiquitous systems.