Aspartame, distributed under several trade names (e.g., NutraSweet® and Equal®), was approved in 1981 by the FDA after numerous tests showed that it did not cause cancer or other adverse effects in laboratory animals (Council on Scientific Affairs 1985; Flamm 1997; Koestner 1997).
In the NIH-AARP Diet and Health Study, aspartame consumption ranged from 0 to 3400 mg per day (about 19 cans of soda at the high end; however, the upper limit is not absolute because investigators asked multiple-choice questions on frequency and the highest option was "6-plus times a day"). There are 180 mg of aspartame in a 12 ounce can of diet soda.
The highest aspartame category in the NIH-AARP Diet and Health Study was "600 mg and above per day," or about three or more cans of diet soda; researchers also examined higher categories (more than 1200 mg per day or 2000 mg per day, which is equivalent to approximately seven to 11 cans of soft drinks daily) with fewer people and found similar results of no elevated risk.
FDA's Acceptable Daily Intake (ADI) of aspartame is 50 mg per kilogram of body weight or about 3,750 mg (21 cans of diet soda) for an adult weighing 75 kilograms (165 lb). ADI is the amount of substance (e.g., food additive) like aspartame that can be consumed daily over a lifetime without appreciable health risk to a person on the basis of all the known facts at the time of the evaluation.
The average aspartame consumption among diet beverage consumers in the NIH-AARP Diet and Health Study was 200 mg per day, or about 7 percent of the ADI, which is the same as a survey of U.S. consumers done by the FDA.
An animal study that fed 0, 4, 20, 100, 500, 2500, and 5000 mg per kilogram of body weight of aspartame to rats saw lymphoma/leukemia increase in female rats, starting from about twice the risk with 20 mg per kilogram of body weight (a person weighing 75 kilograms or 165 lbs, consuming 1500 mg aspartame, or about 8 cans of diet soda) compared with a control group that was not fed aspartame.
This answer is based on source information from the National Cancer Institute.