The potential colour rendering capability and efficiency of white LEDs constructed by a combination of individual red, green and blue (RGB) LEDs are analysed. The conventional measurement of colour rendering quality, the colour rendering index (CRI), is used as well as a recently proposed colour quality scale (CQS), designed to overcome some of the limitations of CRI when narrow-band emitters are being studied. The colour rendering performance is maximised by variation of the peak emission wavelength and relative intensity of the component LEDs, with the constraint that the spectral widths follow those measured in actual devices. The highest CRI achieved is 89.5, corresponding to a CQS value of 79, colour temperature of 3800 K and a luminous efficacy of radiation (LER) of 365 lm/W. By allowing the spectral width of the green LED to vary the CRI can be raised to 90.9, giving values of 82.5 and 370 lm/W for the CQS and LER, respectively. The significance of these values are discussed in terms of optimising the possible performance of RGB LEDs.