For simplicity let us consider the capacitor to be plane parallel, having air as dielectric, with the area of one plate of `S`, and distance between plates `d_0` . Its capacitance is given by
`C_0 = epsilon_0*S/d_0`
When a conducting sheet of a certain thickness `d` is inserted parallel to the plates between them, basically what happens is that the distance between the plates becomes smaller.
`d <d_0 => C > C_0`
Therefore in the case the capacitance increases.
When a nonconducting (dielectric) sheet of a certain thickness `d_1` is inserted parallel to the plates between them, what happens is that two capacitors are formed that are connected in series. First capacitor has as dielectric the sheet, and the second capacitor has the dielectric the air of thickness `d_2`.
`C_1= epsilon_0*epsilon_r*S/d_1` and `C_2=epsilon_0*S/d_2`
with the condition `d_0 =d_1 + d_2`
The total capacitance is
`C = (C_1*C_2)/(C1 + C_2)`
and depends on the nature of dielectric and its thickness. It can be higher or smaller than the capacitance `C_0` of the initial capacitor.