Intracellular transport of large cargoes, such as organelles, vesicles, or large proteins, is a complex dynamical process that involves the interplay of adenosine triphosphate-consuming molecular motors, cytoskeleton filaments, and the viscoelastic cytoplasm. In this work we investigate the motion of pigment organelles (melanosomes) driven by myosin-V motors in Xenopus laevis melanocytes using a high-spatio-temporal resolution tracking technique. By analyzing the obtained trajectories, we show that the melanosomes mean-square displacement undergoes a transition from a subdiffusive to a superdiffusive behavior. A stochastic theoretical model, which explicitly considers the collective action of the molecular motors, is introduced to generalize the interpretation of our data. Starting from a generalized Langevin equation, we derive an analytical expression for the mean square displacement, which also takes into account the experimental noise. By fitting theoretical expressions to experimental data we were able to discriminate the exponents that characterize the passive and active contributions to the dynamics and to estimate the "global" motor forces correctly. Then, our model gives a quantitative description of active transport in living cells with a reduced number of parameters.