A light field camera can be used to make shots which can be truly refocused after the shot, however if I understand the technology correctly it would be just as easy to process the "image data" in such a way that you can get a shot which is entirely in focus, which seems even more interesting than after the fact focusing. However, as far as I know with the current light-field camera(s) this is not possible, so is this technically impossible or is there another reason this has not yet been done? (Such as it being possible somehow with traditional cameras?)
No. They cannot be entirely in focus. That is why there is a focus control.
You see, the lightfield is captured by micro-lenses splitting light. This effectively creates a stack of images at different focus-distances. Software interpolates between these to generate an image which is in focus at any range from the nearest to the furthest focus-plane which is captured.
This lets you have a great depth-of-field but not an infinite one. If you want to cover a much wider range of focus, you will have to resort to manually focus-stacking because you can move and set the lens at a greater number of distances and with greater depth. The difference in focus between micro-lenses is just not that big.
It is certainly possible through a similar principal to focus stacking. If that feature actually isn't available, it's entirely opinion as to why it hasn't been added unless they have some disclosure of that, but the same focus stacking technique that works for traditional images would work for individual slices taken from a light-field shot.
As for why this is necessary with a light-field camera, think about determining what part of the image is in focus. The camera may capture the image at an infinite number of focal depths, but in order to piece together an image that is more in focus than the optical system would have produced for a given selection of focal plane, parts of the image have to be taken from different focal distances and this means that each pixel has to be selected for the most in focus distance and blend them with the sharpest parts from other parts. This is the same if your source is either a light-field camera or a still camera, the only difference is the simplicity in capturing the depth information.
If I had to guess, I'd say it is because the entire point of a light field camera is that it gives depth of field blur but still allows shifting. If someone want's an image entirely in focus, there's not much reason to use a light-field camera since you don't gain anything by its use, but you give up a whole lot of resolution.
That said, I can also see the argument that you might want to simultaneously have an image that could have different things in focus and a copy of the image with everything in focus, so I don't necessarily think that is good reasoning.
Update: It appears that the software is actually capable of doing this already. On this review of the new Lytro, if you do the perspective shift, everything comes in to focus.