Large intelligent surfaces (LISs) constitute a new and promising wireless communication paradigm that relies on the integration of a massive number of antenna elements over the entire surfaces of man-made structures. The LIS concept provides many advantages, such as the capability to provide reliable and space-intensive communications by effectively establishing line-of-sight (LOS) channels. In this paper, the system spectral efficiency (SSE) of an uplink LIS system is asymptotically analyzed under a practical LIS environment with a well-defined uplink frame structure. In order to verify the impact on the SSE of pilot contamination, the SSE of a multi-LIS system is asymptotically studied and a theoretical bound on its performance is derived. Given this performance bound, an optimal pilot training length for multi-LIS systems subjected to pilot contamination is characterized and, subsequently, the performance-maximizing number of devices that the LIS system must service is derived. Simulation results show that the derived analyses are in close agreement with the exact mutual information in presence of a large number of antennas, and the achievable SSE is limited by the effect of pilot contamination and intra/inter-LIS interference through the LOS path, even if the LIS is equipped with an infinite number of antennas. Additionally, the SSE obtained with the proposed pilot training length and number of scheduled devices is shown to reach the one obtained via a brute-force search for the optimal solution.