It’s complicated. Short version, over a small amount of time.
In the case of water, how it kills you is by diluting your blood, basically. Your kidneys will be working extremely hard (and quickly) to empty out the excess water, so for the most part, you’ve got to drink much faster than your kidneys can work.
That said, it’s not just speed - other stuff gets cleaned out with your urine, like certain vitamins and such. Drinking excess water over a long time, but significantly more than what’s on the chart, will drain you of certain nutrients / electrolytes, and that’ll screw you, too.
Drinking sports drinks in that quantity could actually sidestep that particular problem, drastically raising the amount of water you can take in.
One way or another, though, while it’s possible to hurt or kill yourself from drinking too much water, you have to bring it to some serious extremes and your body should be vehemently complaining during this process.
If ever you think you’re doing something extreme and might possibly be slightly risky in this regard, just drink some electrolyte heavy stuff instead for a while - Gatorade, Powerade, etc. Then your only risk is basically outrunning your kidneys and your stomach should really be making you throw up if you try that.
