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This is related to "Is Image Stabilization better in the lens or the body?" - there are lenses with stabilization built-in, and there are bodies with it built-in.
What if a stabilizing lens is combined with a stabilizing body? Would the result be over-compensated, making as bad a mess as the original, or would the camera stabilization attempt to correct any residual motion the lens didn't compensate for?
If the stabilisation in the camera is digital (i.e. analyses the image to do corrections), it would work with a stabilising lens. However, the stabilisation in the camera would only add anything when the movements are too much for the lens to handle on it's own.
If they are both mechanical, they react to movement of the camera instead of movement of the picture, so together they would overcompensate.
Yes, it would be over-compensated: The lens and body stabilization systems don't communicate with each other. You should disable one or the other.
I believe there were more standardized tests done on forums, but here is one.
Until a manufacturer offers such a combination, then we can only hypothesise.
The only case where I could see this being developed/useful, would be if the correction method differed between the two, for example, the lens may correct for vibration, whilst the body could potentially correct rotation.
If both were trying to solve the same type of problem (e.g. vibration), then you may end up with the camera disabling its stabilisation when a stabilised lens attached (or vice versa), so they didn't both try and fix the problem (leading to over-compensation), on the logic that one of them would have "more advanced" stabilisation.
It's certainly plausible that a system could evolve that used both, and it's plausible that it might work better (since they have different strengths). But it would be more expensive, so I doubt anyone would ever build such a system.
I believe you can actually do this (e.g. see Eruditass' answer, and perhaps in the 4/3 cameras?), but as mentioned, if the two systems don't communicate, they will overcompensate.
When properly implemented so that they work together, it works great. Both Sony and Canon (and probably many others) have advanced to the point that both IBIS and IS can be used at the same time to complement the strengths and weaknesses of each.
However, the gain is not purely additive. A camera with, say, five stops of usable IBIS when combined with a lens that offers, say, four stops of usable IS does not increase the total gain to nine stops of usable IS+IBIS. Rather, the total gain may be somewhere around six or seven stops.
