A quick look on Amazon reveals that most cookie cutters are two to four centimetres tall. The ones I have at home vary between two and a half and three centimetres.
As it's a question on how much material you need to finish your cookie cutter, I think that three centimetres are more than enough (two might also be enough). Four seems unreasonably high. Most cookie doughs are rolled until they are very thin so that you don't really need tall cookie cutters.