Can drinking a lot of water be fatal?
Based on what I gathered from this Wikipedia article, Yes.

Drinking copious amounts of water can prove fatal. The proper term is "Water intoxication".

When you start taking in a lot of water (by "a lot" I mean more water than your body can excrete via sweat or urine), the interstitial fluid that bathe the cells that form your (living) tissue end up getting "diluted", i.e- the concentration of ions like $\mathrm{Na^{+}}$ is greatly reduced. The result? The concentration of ions in the interstitial fluid is far lower than the concentration of ions inside the cells it surrounds. Thinking of this in another way; the concentration of water in the interstitial fluid is far higher than the concentration of water inside the cells.

Ever soaked dry raisins in water? Over time they begin to absorb water and swell up as a consequence of the concentration gradient set up between the tissue in the raisin (Low concentration of water) and the water in the bowl. This phenomenon, where water flows from a region of higher concentration (of water) to a region of lower concentration (of water) across a semi-permeable membrane is called osmosis.

It's pretty much the same thing in the case of water intoxication. A concentration gradient has been set up between cells and the interstitial fluid that surrounds it. So water begins to flow into the cell, and the cell swells up quite a bit, resulting in a build-up of turgor pressure

This increase in turgor pressure will be seen to varying degrees in all sorts of (living) tissue, especially tissue that is highly vascular. An important example would be neural tissue, particularly that in the brain. This swelling up of tissue results in an increase in intracranial pressure which can lead to loss of functionality over time. This is what makes water intoxication so lethal.

There isn't really much about the chemistry of water that's responsible for this, apart from osmosis that is.



Addendum:

Drinking a ton of water isn't the only way for water to build up in potentially lethal amounts in the body. Hyponatremia, which refers to a condition of low (below-normal) sodium levels in body fluids, can also lead to water accumulating in the body in excess.