Yes the claim that you can drink too much water is true. Multiple sources, like the Cleveland Clinic, Healthline, and the Mayo Clinic all have articles and research done about overhydration and its legitimacy. Overhydration or Water Intoxication is a real thing and drinking too much water leads to your kidneys to have too much to filter, and they can't get rid of the excess water.
"Water intoxication is a condition in which there’s too much water in your body, and it affects your body’s cells. Mild symptoms include nausea, vomiting and bloating. Severe symptoms may include confusion, seizure and coma. It can also be fatal. Treatment may include reducing your water intake. In severe cases, you may need replacement sodium." (Cleveland Clinic)
"You can become overhydrated in two ways: By drinking too much water or if your kidneys retain too much water. In both cases, overhydration can lead to water toxicity, also known as water poisoning. Your body’s water volume becomes too large for your kidneys to excrete, which can cause the electrolytes in your body to dilute." (Healthline)
"When you drink too much water, your kidneys can't get rid of the excess water. The sodium content of your blood becomes diluted." (Mayo Clinic)
https://my.clevelandclinic.org/health/diseases/water-intoxication
https://www.healthline.com/health/overhydration#_noHeaderPrefixedContent
https://www.mayoclinic.org/healthy-lifestyle/nutrition-and-healthy-eating/in-depth/water/art-20044256