As more and more users access social network services from smart devices with GPS receivers, the available amount of geo-tagged information makes repeating classical experiments possible on global scales and with unprecedented precision. Inspired by the original experiments of Milgram, we simulated message routing within a representative sub-graph of the network of Twitter users with about 6 million geo-located nodes and 122 million edges. We picked pairs of users from two distant metropolitan areas and tried to find a route between them using local geographic information only; our method was to forward messages to a friend living closest to the target. We found that the examined network is navigable on large scales, but navigability breaks down at the city scale and the network becomes unnavigable on intra-city distances. This means that messages usually arrived to the close proximity of the target in only 3-6 steps, but only in about 20% of the cases was it possible to find a route all the way to the recipient, in spite of the network being connected. This phenomenon is supported by the distribution of link lengths; on larger scales the distribution behaves approximately as P(d) ≈ 1/d, which was found earlier by Kleinberg to allow efficient navigation, while on smaller scales, a fractal structure becomes apparent. The intra-city correlation dimension of the network was found to be D2 = 1.25, less than the dimension D2 = 1.78 of the distribution of the population.