In this paper, we analyze the effect of channel estimation errors on the performance of an uplink cellular network. We use a stochastic geometric approach, where the Mobile Users (MUs) and the Base Stations (BSs) are modeled as being randomly located on the 2-dimensional plane according to independent Poisson point processes. Each MU makes use of fractional distance-dependent power control, while transmitting both the training signal as well as the data signal in the uplink direction. We derive an analytical expression for the uplink coverage probability for a typical BS-MU pair, accounting for the effects of channel estimation errors and fractional power control. We numerically obtain the fractional power control that maximizes the coverage probability, and show that the optimal power control factor does not depend on the training duration. Further, we numerically compute the optimal training duration that maximizes the area spectral efficiency. The results provide critical insights into the design and optimization of uplink cellular networks in the presence of pilot contamination due to practical channel estimation.