Kelvin temperature refers directly to the color of the light coming out of the lamp. The Kelvin does not change or affect the amount of light (lumens) being released from the bulb, nor does it directly correlated to the Color Rendering Index (CRI) of the bulb. Kelvin temperature simply refers to the color temperature you see when looking at or around the bulb.
Kelvin temperatures typically range from 2700K to 6500K. 2700K would be closest to the orange light seen around a High Pressure Sodium lamp. This type of temperature is considered warm. Consumers might be inclined to think that this temperature has less light (lumens) than a higher kelvin lamp, they would be wrong. The mean lumens out of the fixture will be the same for both temperatures. The only difference would be the color of the light being emitted. The naked human eye might consider that a white light similar to daylight would be brighter, however it is simply more white.
An example of a whiter light would be 6000K which is more to what you would expect from an LED lamp. 6500K would be an even brighter white, bordering on having a bluish hue. In terms of CRI it is true that a 2700K bulb would have lower CRI, but this is because it is falling into such the warm orange spectrum that it is affecting the true rendering of colors you are seeing. The difference, however, between 5000K and 5700K in terms of CRI would be non existent.