Yes, it is possible to 'drink too much water' - but it would be more accurate to state that it's not the water that is a problem, but rather the electrolytes. This is a condition called "Water Intoxication" and is something that happens to athletes - in particular long distance runners. Imbalances in electrolytes result when water is consumed to replace what is lost but lost salts (like sodium/potassium) are not also replaced.

You can read more about water intoxication on this site by clicking the link provided, you can also try a search on this site using the icon located to the right of every page.

HTH
Nat