Large MIMO transceivers are integral components of next-generation wireless networks. However, for such systems to be practical, their channel estimation process needs to be fast and reliable. Although several solutions for fast estimation of sparse channels do exist, there is still a gap in understanding the fundamental limits governing this problem. Specifically, we need to better understand the lower bound on the number of measurements under which accurate channel estimates can be obtained. This work bridges that knowledge gap by deriving a tight asymptotic lower bound on the number of measurements. This not only helps develop a better understanding for the sparse MIMO channel estimation problem, but it also provides a benchmark for evaluating current and future solutions.