In the near future, it is envisioned that cellular networks will have to cope with extensive internet of things (IoT) devices. Therefore, a required feature of cellular IoT will be the capability to serve simultaneously a large number of devices with heterogeneous demands and qualities of channel state information at the transmitter (CSIT). In this paper, we focus on an overloaded multiple-input single-output (MISO) broadcast channel (BC) with two groups of CSIT qualities, namely, one group of users (representative of high-end devices) for which the transmitter has partial knowledge of the CSI, the other group of users (representative of IoT devices) for which the transmitter only has knowledge of the statistical CSI (i.e., the distribution information of the user channels). We introduce rate-splitting multiple access (RSMA), a new multiple access based on multi-antenna rate-splitting (RS) technique for cellular IoT. Two strategies are proposed, namely, time partitioning-RSMA (TP-RSMA) and power partitioning-RSMA (PP-RSMA). The former independently serves the two groups of users over orthogonal time slots while the latter jointly serves the two groups of users within the same time slot in a non-orthogonal manner. We first show at high signal-to-noise ratio (SNR) that PP-RSMA achieves the optimal degrees-of-freedom (DoF) in an overloaded MISO BC with heterogeneous CSIT qualities. We then show at finite SNR that PP-RSMA achieves explicit sum rate gain over TP-RSMA and all baseline schemes by marrying the benefits of PP and RSMA. Furthermore, PP-RSMA is robust to CSIT inaccuracy and flexible to cope with quality of service (QoS) rate constraints of all users. The DoF and rate analysis helps us in drawing the conclusion that PP-RSMA is a powerful framework for cellular IoT with a large number of devices.