In order to trigger an adaptive immune response, T cells move through lymph nodes searching for dendritic cells that carry antigens indicative of infection. We observe T cell movement in lymph nodes and implement those movement patterns as a search strategy in a team of simulated robots. We find that the distribution of step-sizes taken by T cells are best described by heavy-tailed (Lévy-like) distributions. Such distributions are characterized by many small steps and rare large steps. Our simulations show that heavy-tailed motion leads to dramatically faster search compared to Brownian motion, both in groups of T cells and in teams of robots. The mechanisms that cause heavy-tailed movement patterns in T cells are not fully understood. However, in robot simulations we find that heavy-tailed movement improves search speed whether that movement is caused by rules intrinsic to the robots or by adaptive response to extrinsic factors in the environment.