Complement of answer
A1 This can be possible
A2. This can't be possible.
As explained in user Edwin Ashworth it seems that given a plain possibility, this construction refers twice to the same possibility (or, as well, that one of two possibilities is missing in the context) thereby the sense of imprecision it communicates in a simple context. However, in the context of a possibility subordinate to another, it seems also that this is a sound construction.
— Could it be that the wind banged the door shut?
— This can become a possibility only if there was a sufficiently strong wind that day, and if the corridor pulls in strong enough draughts.
— (After two phone calls) This can be possible: it was windy and the caretaker confirms that the wind in the corridor can at times bang the doors shut.
A negative alternative in this dialogue is in fact naturally put into words by means of the negative (A2).
— This can't be possible; there was no wind that day.
B Can it be possible to refine this further?
The same comment as in "1." can be made for "B", insofar as the felt impropriety. As to assert that complex contexts justify its use, I have no certitude. It seems that in a context where the "final" possibility is subordinate to the basic possibility of being a particular case, some other construction might be used, as the following, for instance.
ex.: possibilities of application of a processus of restauration to various paintings (hypothetical)
— Is it ever possible to refine this further?
— Yes, it is, but in rare cases when the painting is not too old and if you can find an expert willing to take the risk.