He's not wrong, matter of quantity: https://en.wikipedia.org/wiki/Water_intoxication

('Notable cases' include recorded death.)

Well obviously any lifeform has water problems. Like any other lifeform, our life depends on remaining within osmotic balance with whatever we drink and eat.

That said, the amount of ways your body adapts to osmotic balance problems is in the dozens, so poisoning your body with water, while possible, is hard to the tune of it requires something like drinking 20 liters of water in less than an hour. I mean, it seems to be possible but I cannot imagine succeeding at that even if I'm trying. That's 2 full buckets (and that's just LD50, as in, if you're >70kg it's more than that).

There is an exception: purified water. You cannot drink mineral-poor* or distillated water safely in even small quantities. Why not ? Because that will locally, suddenly and very strongly disrupt the osmotic balance in what you bring it into contact with. It probably not kill you (not without repeated application), but it may cause serious issues in your mouth, throat and stomach. It should also not be used to make coffee or tea and drunk, that merely lessens the effect it doesn't stop it.

* mineral-poor water is something used in chemical and medical laboratories

It may be hard to imagine, but it has happened: http://articles.latimes.com/2007/jan/14/local/me-water14

I was thinking of things like fluoride, water chlorination, etc. It's also possible to drink water that is "too filtered" and doesn't have enough minerals in it which can apparently cause issues.

Yeah but that's way way past the point where it's very uncomfortable to drink more water. It's not something that can happen to you without warning (or while you're enjoying the process).

Anecdote: My mother got quite sick while drinking a huge amount of water in preparation for a religious 26 hour fast.

Was she comfortable drinking that amount, or was she forcing herself to?