We study information-theoretic limits for simultaneous wireless information and power transfer (SWIPT) systems employing nonlinear radio frequency (RF) energy harvesting (EH) receivers (Rxs). In particular, we consider a SWIPT system with one transmitter that broadcasts a common signal to an information decoding (ID) Rx and multiple EH Rxs. Owing to the nonlinearity of the EH Rxs' circuitry, the efficiency of wireless power transfer depends on the waveform of the transmitted signal. We aim to answer the following fundamental question: What is the optimal input distribution of the transmit signal waveform that maximizes the information transfer rate at the ID Rx conditioned on individual minimum required direct-current (DC) powers to be harvested at the EH Rxs? Specifically, we study the conditional capacity problem of a SWIPT system impaired by additive white Gaussian noise subject to average-power (AP) and peak power (PP) constraints at the transmitter and nonlinear EH constraints at the EH Rxs. To this end, we develop a novel nonlinear EH model that captures the saturation of the harvested DC power by taking into account the reverse breakdown current of the rectifying diode. We derive a novel semiclosed-form expression for the harvested DC power, which simplifies to closed form for low input RF powers. The derived analytical expressions are shown to closely match circuit simulation results. We solve the conditional capacity problem for real- and complex-valued signalling and prove that the optimal input distribution that maximizes the rate-energy (R-E) region is unique and discrete with a finite number of mass points. Furthermore, we show that the boundary of the R-E region saturates for high PP constraints due to the saturation of the harvested DC power for high input RF powers. In addition, we devise a suboptimal input distribution whose R-E tradeoff performance is close to optimal.