FDA acceptable daily intake (ADI) for aspartame is 50 mg per kg of body mass.[0] For an individual 180 pounds, that's about 82 kg. That means his ADI is 4100 mg. Aspartame in popular diet sodas is between 50 and 125 mg.[1]
You'd have to drink A LOT of diet soda to reach these levels.
I have a friend who used to drink nothing but Diet Coke. He went through a few 2l bottles a day. According to that link (assuming it's the same in Europe), Diet Coke is 125mg per 8 ounces, so a 2l bottle is 125 / 8 x 70.4 = 1100mg. So if he drank four 2l bottles, that's over the daily limit.
You'd have to drink A LOT of diet soda to reach these levels.
[0] http://www.cancer.org/cancer/cancercauses/othercarcinogens/a...
[1] http://static.diabetesselfmanagement.com/pdfs/DSM0310_012.pd...