Can you mix satin finish paint with eggshell finish paint?

More important is, that both paints are of the same kind (like: oil-based paint, acrylic paint, latex paint). Semi-gloss is more easily painted over then high gloss, but first better find out what the old paint layer is made of. Check with the help of alcohol - if paint comes off in any way when you rub it with a cloth of alcohol, it's latex or acrylic paint (and can be painted over with acrylics).