Many self-diffusion measurements have been performed on giant micelles by either fluorescence recovery after photobleaching (FRAP) or Fourier transform pulsed-gradient spin echo (FT PGSE) and reported by many authors. As a general feature, one observes a minimum of the diffusion coefficient as a function of surfactant concentration, which is not clearly understood and has often been attributed to the existence of connections. In this paper, we explain that this minimum may result from the competition between two mechanisms : the diffusion of the micelle itself and the surfactant molecule diffusion on the micelle.