Covert communications can hide the very existence of wireless transmissions and thus are able to address privacy issues in numerous applications of the emerging Internet of Things (IoT). In this article, we adopt channel inversion power control (CIPC) to achieve covert communications in Rayleigh fading wireless networks, where a transmitter can possibly hide itself from a warden while transmitting information to a receiver. The CIPC can guarantee a constant signal power at the receiver, which removes the requirement that the receiver has to know the channel state information in order to coherently decode the transmitter's signal. Specifically, we examine the performance of the achieved covert communications in terms of the effective covert rate (ECR), which quantifies the amount of information that the transmitter can reliably convey to the receiver subject to the warden's total error probability being no less than some specific value. The noise uncertainty at the warden serves as the enabler of covert communications. For fairness, we also consider noise uncertainty at the legitimate receiver. Our examination shows that increasing the noise uncertainty at the warden and the receiver simultaneously may not continuously improve the ECR achieved in the considered system model.