The kinetics of loop formation, i.e., the occurrence of contact between two atoms of a polypeptide, remains the focus of continuing interest. One of the reasons is that contact formation is the elementary event underlying processes such as folding and binding. More importantly, it is experimentally measurable and can be predicted theoretically for ideal polymers. Deviations from single exponential kinetics have sometimes been interpreted as a signature of rugged, protein-like, free energy landscapes. Here we present simulations, with different atomistic models, of short peptides with varied structural propensity, and of a structured protein. Results show exponential contact formation kinetics (or relaxation) at long times, and a power law relaxation at very short times. At intermediate times, a deviation from either power law or simple exponential kinetics is observed that appears to be characteristic of polypeptides with either specific or nonspecific attractive interactions but disappears if attractive interactions are absent. Our results agree with recent experimental measurements on peptides and proteins and offer a comprehensive interpretation for them.