According to researchers and medical professionals drinking too much milk can cause more health issues such as with your digestive health, make your bones more brittle and can increase your risk of early death.
However you really would have to drink a whole lot of milk before it actually became a problem and got that severe.
Drinking a few glasses of milk per day is not gonna hurt you and will actually make you healthier and make your bones stronger.
If you do start drinking too much milk then you'll likely have a stomach ache, stomach cramps, bloating or diarrhea.
If so then you should slow down on the amount of milk you're drinking.