When biological cells migrate, divide, and invade, they push and pull on individual fibers of the matrix surrounding them. The resulting fiber displacements are neither uniform nor smooth; rather, displacements localize to form dense fibrous bands that span from one cell to another. It is thought that these bands may be a mechanism by which cells can sense their neighbors, but this hypothesis remains untested, because the mechanism for band formation remains unknown. Using digital volume correlation, we measure the displacements induced by contractile cells embedded in a fibrous matrix. We find that cell-induced displacements propagate over a longer range than predicted by linear elasticity. To explain the long-range propagation of displacements, we consider the effect of buckling of individual matrix fibers, which generates a nonlinear stress-strain relationship. We show that fiber buckling is the mechanism that causes the displacements to propagate over a long range and the bands to form between nearby cells. The results thus show that buckling of individual fibers provides a mechanism by which cells may sense their distant neighbors mechanically.