Yes, you can drink the water in Chile. But not all sources are the same...
Let's start in Santiago. I don't know of any Gringo that has ever had a problem with tap water in Santiago – in recent years anyway. I am sure it has happened at some point; but I encourage people to drink tap water and it's not a problem. In restaurants, your server will encourage you to drink bottled water. If you want to drink tap water, ask for "agua de la llave." You'll appear to know what's up and they won't hound you.
As in the rural parts of the U.S., when you get out of the cities, there are a variety of systems for collecting water, from municipal systems to wells to run-off collecting and even pipes into streams. For the most part, I find the water quality in Chile to be excellent. In the mountains, I often drink directly from the streams.
Using common-sense is advised. If you see that the pipe is being fed by a stream with a cow standing in it, well, you might not want to drink that water. I know some of my guests have more sensitive systems and get very mildly irritated – just like they can when they visit Colorado. My advice is try the water for a day or two, if your system changes, and you become uncomfortable, try bottled water and see if it improves.