Is it possible for a nonnormal magic square to have the same magic constant as a normal magic square of the same order?

Question

I've been programming some quick and fast solutions to 4x4 magic square problems, and I've come across an assumption I've been using: The only squares with the magic constant 34 are the normal 4x4 magic squares.

Is this assumption correct? Can it be generalized to magic squares of any order?