When scientists were testing the effects of aspartame on rats and mice and things of the like, they fed them such massive amounts of aspartame that they got quite ill. If you feed a rat that much of anything it'll make them sick. The scientific research there was shady, at best.
Some people can actually have allergies to aspartame, just like you can have allergies to milk or peanuts. But as far as it ever causing cancer... unless you eat several times your body weight in aspartame every week, I think you're safe
