With the severe spectrum shortage in conventional cellular bands, millimeter wave (mmW) frequencies between 30 and 300 GHz have been attracting considerable attention as a possible candidate for next-generation micro- and picocellular wireless networks. The mmW frequency bands offer orders of magnitude greater spectrum than current cellular microwave frequencies currently deployed below 3 GHz. However, even with typical microcellular radii of 100m to 200m, the propagation of mmW signals in outdoor non line-of-sight (NLOS) links remains challenging and the feasibility of such mmW networks is far from clear. This paper uses recent real-world measurements at 28 GHz to provide the first systematic assessment of mmW picocellular networks. It is found that, even with its limited propagation characteristics, mmW systems can offer an order of magnitude increase in capacity over current state-of-the-art 4G cellular networks with similar cell density. However, it is also shown that such mmW networks will operate in an extremely power-limited regime where the full spatial and bandwidth degrees of freedom are not fully utilized. This power-limited regime contrasts significantly with current bandwidth-limited cellular systems, requiring alternate technologies for mmW systems that may unlock further gains that mmW frequency bands offer.