The distance estimation problem is investigated for visible light positioning (VLP) systems with red-green-blue (RGB) light emitting diodes (LEDs). The accuracy limits on distance estimation are calculated in terms of the Cramér-Rao lower bounds (CRLBs) for three different scenarios. Scenario 1 and Scenario 2 correspond to synchronous and asynchronous systems, respectively, with known channel attenuation formulas at the receiver. In Scenario 3, a synchronous systems is considered but channel attenuation formulas are not known at the receiver. The derived CRLB expressions reveal the relations among the distance estimation accuracies in the considered scenarios and provide intuitive explanations for the benefits of using RGB LEDs.