Joseph Rao asks about problems with isolated* capacitors. They usually boil down to using semi-quantitative reasoning with Q = CV.
* Isolated capacitors, as opposed to capacitors in circuits with resistors.
When the capacitor remains connected to a battery, the voltage across the capacitor cannot change. Thus any structural change in the capacitor which changes capacitance will also change the charge stored on the capacitor. For example, doubling the distance between parallel plates halves the capacitance (by C = εA/d). Then with voltage constant, Q = CV tells us the charge stored has also halved.
When the capacitor is not connected to any battery or external circuit, then the plates are insulated; the charge stored on those plates cannot change. In this case, any capacitance change will change the voltage across the capacitor. For example, doubling the distance between parallel plates again halves the capacitance; but if the capacitor isn't connected to a battery, the charge on the plates won't change, meaning that the voltage across the capacitor doubles by Q = CV.
Capacitance is a property of the capacitor's structure. Just changing the voltage across or charge stored on a capacitor does not affect capacitance. If we don't change the area of the plates, the distance between the plates, or the dielectric material in between the plates, the capacitance will remain unchanged. In this case, doubling the voltage across the capacitor would double the charge stored by Q = CV at constant C.