Ummmm.....am terribly sorry to be a bearer of bad news (and I excuse anyone with PKU from this category), but it's not really likely for most of us to be intolerant of aspartame.
Aspartame is a dipeptide, made of two amino acids, phenylalanine and aspartic acid. As amino acids are the building blocks of proteins, which are completely necessary to our metabolic goings-on, intolerance is rare. People with the inborn error of metabolism, Phenylketonuria, cannot metabolise phenylalanine, and so have higher than normal serum and/or tissue concentrations of the amino acid, with eventual physical symptoms and sequelae. PKU is screened for just after birth, and if the initial blood test (from a heel stick) shows a possibility of the disease, further tests are done. It's treated with a low protein (basically) diet; if this is followed, the person with PKU can attain normal development and lifespan.
Studies suggest that some people are more sensitive to the slight bitterness in aspartame, and view this as intolerance.