Is it possible to drink too much water while pregnant?

Yes. Water intoxication is rare but deadly. Just follow your body's clues. Sensations of hunger and thirst are our body's way of telling us what it needs. I never tell pregnant women to force themselves to drink water. Drink to quench your thirst.
Possible. Drinking too much water can be dangerous in any person causing electrolyte imbalances, but it has to be a lot. Discuss with your gyn the proper amount of fluids including water that you should drink at your weight/height and during your pregnancy. Good luck.