Excess water enters blood
Water absorbed from the gut enters circulation and dilutes blood components, including sodium. The danger is usually rate plus volume, not ordinary sipping.
Body & Brain
Yes, you can drink too much water. And the result is stranger and more dangerous than you'd expect. In 2007, Jennifer Strange died after a Sacramento radio contest called Hold Your Wee for a Wii, in which contestants drank large amounts of water without urinating. Water feels synonymous with health, but the body's relationship with water is about balance, not maximum intake. Drinking too much water does not drown you. It can kill you by diluting blood sodium until brain cells swell inside the skull.
Quick answer
Drinking too much water too quickly can cause hyponatremia, dangerously low blood sodium. As sodium falls, water moves into cells by osmosis, including brain cells. Brain swelling can cause headache, confusion, seizures, brain herniation, and death. Healthy kidneys can process roughly 0.8-1 liter per hour. Many sports drinks were developed not just to replace water, but to replace electrolytes, because plain-water overdrinking can make endurance athletes sick.

The short answer
Drinking too much water too quickly can cause hyponatremia, dangerously low blood sodium.
Excess water enters blood
Water absorbed from the gut enters circulation and dilutes blood components, including sodium.
Curiosity twist
Many sports drinks were developed not just to replace water, but to replace electrolytes, because plain-water overdrinking can make endurance athletes sick.
Common mistake
Excess water is harmless because the kidneys will simply excrete it.
Next tiny mystery

Body & Brain
Another big-question explanation in the same collection.

Body & Brain
Another big-question explanation in the same collection.

Body & Brain
Another big-question explanation in the same collection.