Ever since Neil DeGrasse Tyson mentioned the "lead in the water" theory of why Rome fell on Cosmos, scientists and historians have been tearing up the internet to explain all the problems with this idea.
Tyson described the way Romans preferred to use lead pipes for their drinking water, as well as other eating and drinking vessels. He noted that "many historians" think the Romans' plumbing habits could have lead to "a mass poisoning on an unprecedented scale" that precipitated the great Empire's decline.
A paper published in Proceedings of the National Academy of Sciences a few days before the Cosmos episode aired confirmed one part of Tyson's statement. Researchers conducted careful analysis of sediment cores from a harbor that once held ancient Rome's water runoff. What they discovered was that yes, the Romans were likely drinking water that had "100 times more lead than local spring waters."
Would that have constituted a "mass poisoning"? One of the lead researchers told The Guardian's Ian Sample that he was dubious:
Albarède believes that any health problems caused by lead piping could not have brought the civilisation to its knees.
"Can you really poison an entire civilisation with lead? I think it would take more than lead piping in Rome to do that," he said.
Perhaps the most interesting discussion was on Reddit's r/askhistorians, where several participants posted relevant evidence on both sides of the debate. There, Talleyrayand points out that this idea goes back to a book published in 1983, called Lead and Lead Poisoning in Antiquity. Its author, a chemist named Jerome Nriagu, was probably the first to claim lead poisoning precipitated the Roman Empire's fall. There's little evidence to back up his claims, but Talleyrayand notes that historians find significant evidence that the Romans were aware of lead poisoning and took pains to avoid it.
Finally, and perhaps most importantly, even if you accept the idea that lead poisoning leads to elevated crime rates, it's still not the "reason" why Rome fell. There is no single, simple reason this political entity slowly disintegrated after existing for so many centuries in so many different forms.
Blaming Rome's centuries-long decline on lead poisoning is a bit like blaming the fall of the British Empire on liver poisoning from all those gin and tonics downed by the colonists.