It'd be 100% possible with an RGB LED, or a future full-spectrum LED, but not with any of our current "white" LEDs.
Actually, the easiest and currently most effective way to have variable color temperature at the moment would be with two current "white" LEDs, NOT RGB.
RGB is actually surprisingly bad as far as CRI goes. You need yellow wavelengths in between the red and green "spikes" to make it truly usable as a flashlight. While the brain interprets a mixture of red and green as yellow, the problem is that pigments, or objects in the real world don't necessarily
reflect a combination of red and green the same as they would a true broad spectrum light source (or even a good approximation of one, like a decent phosphor LED). For example, wood under RGB light tends to look ORANGE, not brown -- because it reflects a lot of red and very little green. Adding a yellow-wavelength LED to the mix however tends to fix this up nicely. The problem is, balancing and mixing the colors would be hard. Especially considering the red and yellow have different responses to temperature than green and blue, some sort of temp probe + lookup table to determine the currents, or even optical feedback, would necessary to keep the white balance. You also get "rainbow fringes" when tryng to merge multi-color LEDs into "white".
IMO The best way to do soemthing like a variable CCT light would be to simply get two different LEDs -- one cool white (such as a WC bin Cree), and another that is very warm white (such as a 7A Cree), and control the ratios between them. Because both are effectively "white", any combination of the two will also be white of an intermediate color temp. Much easier than trying to mix RYGB primary colors in the right ratios. Mixing multiple different phosphor white LEDs together also tends to "average out" minor differences due to the "tint lottery".
I woudln't mind seeing something like a Cree MC-E (which has 4 separately addressable dice) come out, with half of the dice coated in warm-white phosphor, and the others in cool white (diagonally opposed would be the easiest to blend optically). As there are times where one emitter is simply easier/more elegant to work with -- even if that means de-focusing it or sacrificing throw. That said, uneven beam mixing is far less objectionalbe with two different color of "white" than it is with entirely different colors, like RGB. For example, doing work in an office with both icnan and fluorescent lights on at the same time is not too bad. Trying to work under say theatrical lighting with purple, blue, green etc. shadows all over the place would be worse.
