We consider the uplink (UL) and downlink (DL) of non-cooperative multi-cellular time-division duplexing (TDD) systems, assuming that the number N of antennas per base station (BS) and the number K of user terminals (UTs) per cell are large. Our system model accounts for channel estimation, pilot contamination, and an arbitrary path loss and antenna correlation for each link. We derive approximations of achievable rates with several linear precoders and detectors which are proven to be asymptotically tight, but accurate for realistic system dimensions, as shown by simulations. It is known from previous work assuming uncorrelated channels, that as N→∞ while K is fixed, the system performance is limited by pilot contamination, the simplest precoders/detectors, i.e., eigenbeamforming (BF) and matched filter (MF), are optimal, and the transmit power can be made arbitrarily small. We analyze to which extent these conclusions hold in the more realistic setting where N is not extremely large compared to K. In particular, we derive how many antennas per UT are needed to achieve η% of the ultimate performance limit with infinitely many antennas and how many more antennas are needed with MF and BF to achieve the performance of minimum mean-square error (MMSE) detection and regularized zero-forcing (RZF), respectively.