Integrated single-photon sources are a key component for photonic quantum technology but are generally limited to low single-photon rates. For sources based on photon pair generation by four-wave mixing, increasing the repetition rate of pump laser pulses is a straightforward way to enhance the single-photon rate, but the benefits and practical limitations have not yet been demonstrated and analyzed in a CMOS-compatible platform. In this work, we demonstrate correlated photon pair generation in integrated silicon nanowires and systematically analyze the count rate and coincidence to accidental ratio as the pump rate is varied between 156.25 MHz and 10 GHz. We show that the highest useful pump rate is limited by the timing resolution of the single-photon detection system, and that in this regime, the nonlinear loss of the silicon nanowire does not have a significant effect on the single-photon generation.