Yes, it is possible to drink too much water. You linked an article from WebMD, which is a for-profit agency founded in 1998 that has published thousands of digital articles on health and nutrition. Healthcare professionals vet the information contained in WebMD articles, but WebMD does not claim to be an accredited medical agency (WebMD, Media Bias/Fact Check). The WebMD article that you linked claims that overhydration can lead to serious medical issues, including hyponatremia, which is a deficiency of sodium in the blood (What Happens, WebMD). This claim is reinforced by another article published by the Mayo Clinic, a non-profit medical group that has been operating for almost 150 years. The article states that hyponatremia can be caused by a number of factors, "ranging from an underlying medical condition to drinking too much water — cause the sodium in your body to become diluted" (Hyponatremia, Mayo Clinic).
Both articles thereafter claim that hyponatremia can cause body water levels to rise and cellular swelling, which can lead to life-threatening health problems, such as brain swelling, which can cause a coma or death. These effects are also highlighted in an article by the National Kidney Foundation, which has been researching and advocating for kidney health for 75 years. The article states that more serious symptoms can occur in more severe cases of hyponatremia, including altered mental status, hallucinations, seizures, coma, and death (Hyponatremia, NKF).
Ultimately, the claim that it is possible to drink too much water is a true statement. Excessive water consumption can lead to serious health complications due to a sodium deficiency in the blood, as stated in the WebMD article and both the NKF and Mayo Clinic articles.
Sources:
What Happens When You Drink Too Much Water?
Hyponatremia - Symptoms and causes - Mayo Clinic
Hyponatremia - Symptoms, causes, treatment | National Kidney Foundation
WebMD - Bias and Credibility - Media Bias/Fact Check