In this article, we provide a much-needed study of UAV cellular communications, focusing on the rates achievable for the UAV downlink command and control (C&C) channel. For this key performance indicator, we perform a realistic comparison between existing deployments operating in single-user mode and next-generation multi-user massive MIMO systems. We find that in single-user deployments under heavy data traffic, UAVs flying at 50 m, 150 m, and 300 m achieve the C&C target rate of 100 kbps -- as set by the 3GPP -- in a mere 35%, 2%, and 1% of the cases, respectively. Owing to mitigated interference, a stronger carrier signal, and a spatial multiplexing gain, massive MIMO time division duplex systems can dramatically increase such probability. Indeed, we show that for UAV heights up to 300 m the target rate is met with massive MIMO in 74% and 96% of the cases with and without uplink pilot reuse for channel state information (CSI) acquisition, respectively. On the other hand, the presence of UAVs can significantly degrade the performance of ground users, whose pilot signals are vulnerable to UAV-generated contamination and require protection through uplink power control.