These companies wouldn't continue to market the product if these claims could be proved.
The manufacturers of aspartame, and the US Food and Drug Administration, insisted for years that aspartame did not break down into toxic byproducts during shipping and storage.
An eleven-year-old sixth-grader, Jennifer Cohen, took up that challenge for a school science project. She designed a clean, simple experimental protocol which shows that, in fact, aspartame does degrade in the container into formaldehyde and diketopiperazine. The older the sample and the higher the ambient temperature of storage, the worse the contamination became. And she paid out of her babysitting earnings to have a professional analytical lab confirm her findings.
I am by no means a sandal-wearing organic food fetishist. I eat red meat and drink black coffee and malt whiskey. I even -- horrors! -- still smoke an occasional tobacco cigarette. I demur from much of the hysteria about supposed chemical and radiological hazards about which there isn't any substantive science on which to base rational fear.
But I stay away from aspartame.
Before having had occasion to read up on things like the Cohen study, I had a spell of involuntary hand tremor and migraine headaches. It baffled my internist, who ordered a bunch of inconclusive studies. I had never connected it to aspartame, though a previous occupant of my desk had left behind in a drawer a box of aspartame sachets of which I had been availing myself several times per week. That would have been my only route of exposure: I don't drink cola products nor do I consume sweetened packaged foods.
That box ran out and, thinking that one artificial sweetener was as good or as bad as another, I replaced it with a saccharine sweetener product. Within three weeks, my hands were steady again and I have not had a migraine since.
By the way, the Snopes link quotes someone who quotes Russell Blaylock. For those who actually prefer to dig into the science, both pro and con, behind any controversial question, one could do worse than to read Dr. Blaylock on the aspartame issue.