It's all to do with the conductivity of the material, the thickness is only a secondary effect because as you reduce the thickness you reduce the conductivity.
The only time that thickness has a direct effect is if you make a surface from layers of a transparent material which are a precise fraction of a wavelength in order to use interference effects to increase or decrease reflectivity.
The details are little different for light and radio frequencies but you can calculate the 'depth' the waves penetrate into the metal quite easily - skin effect