Yes on a caldwell rest. It is just a simple tripod looking rest that sits on a heavy shooting table.
Bamabow
Is the rest surface soft or hard? Do you rest the butt of the gun or hold it in your hands? The amount of "load" you put on the rest can affect the POI as much as the grip. This can be magnified by the firmness of the rest surface. Also, often folks who rest a gun don't keep consistant grip tension as they are not holding the entire weight of the gun in their hands.
I have a front rest that consists of a chunk of truck tire inner tube filled with styrofoam packing peanuts and the ends tied off with hose clamps. I hold the gun in my hands with my wrists setting on a rear bag. When I "rest" a revolver I hold it as if I were going to shoot off-hand and slowly and ever so lightly lower the barrel/frame onto the front bag, just to steady the wobble a bit. I don't put pressure on the bag, which keeps my POI consistant and unchanged relative to shooting off-hand.
Consistancy is the key, and it applies to every part of the equation.