Location and time information about individuals can be captured through GPS devices, GSM phones, RFID tag readers, and by other similar means. Such data can be pre-processed to obtain trajectories which are sequences of spatio-temporal data points belonging to a moving object. Recently, advanced data mining techniques have been developed for extracting patterns from moving object trajectories to enable applications such as city traffic planning, identification of evacuation routes, trend detection, and many more. However, when special care is not taken, trajectories of individuals may also pose serious privacy risks even after they are de-identified or mapped into other forms. In this paper, we show that an unknown private trajectory can be re-constructed from knowledge of its properties released for data mining, which at first glance may not seem to pose any privacy threats. In particular, we propose a technique to demonstrate how private trajectories can be re-constructed from knowledge of their distances to a bounded set of known trajectories. Experiments performed on real data sets show that the number of known samples is surprisingly smaller than the actual theoretical bounds.