Yes, you can in fact drink too much water. Through my research I have found that drinking too much water can lead to something called Hyponatremia water intoxication. It happens when too much water enters your body and causes a dilution of electrolytes and an oversaturation of water in you blood. This leads to swollen cells, which puts pressure on you body, primarily your brain. This can lead to confusion, irritability, drowsiness, headaches, muscle weakness, vomiting, and in rare cases can even lead to death.
To avoid Hyponatremia, it is reccommended to stop drinking water when you no longer feel thirsty. If you find yourself in a situation where you are dealing with Hyponatremia, it is reccomended that you seek immediate medical attention.
According to several reliable sources, including Cleaveland Clinic, WebMD, the National Library of Medicine, and Mayo Clinic, this claim is true
Cleaveland Clinic
WebMD
NCBI
Mayo Clinic