There's this funny "heuristic" converse of the central limit theorem (by Mandelbrot, I think):

The ONLY variables that are normally distributed are those that are averages of many independent, identically distributed variables of finite variance.

Thus, if you cannot find the finite-variance variables that average up to form a variable X, then X is not normally distributed.

(The parts about independent and identically distributed are technical red-herrings. The only essential condition is finite variance.)