Do the cities of Milan and Bergamo have drinkable tap water, or is the practice to buy bottled water?
Tap water is always drinkable in Italy, especially in the cities (of course, it may be a little bit different if you are talking about a cottage in the middle of nowhere).
The real question is how good is that. For instance, in Rome the water has a very strong percentage of limestone and it taste doesn't very good. For this reason, several people tends to buy a water purifier or a simple purifier container like the following one.
In smaller cities, like Bergamo, pipe water is definitely better than bigger cities. Specifically, in Bergamo the pipe water seems to be one of the best in Italy.
Talking about Milano, and big cities in general, it depends on where you live (how far from the main water collector, the quality of the pipes, etc). My suggestion is to either try, ask the neighbours or use a water quality test to check the quality of the water in the apartment.
In any case, it's potable. So you can drink it, or use it for a tea or coffee. It may not taste extremely well or the amount of mineral salt may be greater than what you would expect.