I heard it as:

3 minutes without air
3 days without water
3 weeks with food.

There are accounts of people trapped in building rubble after earthquakes who survived for longer than 3 days without water. There are tales of people who fast or hunger strike for more than 3 weeks. As you noted, this is a rule of thumb. Generally speaking, these are good time frames to keep in mind.

Once you pass the magic barrier, your chance of survival diminishes greatly. As you approach the barriers, your bodies ability to anything other than lay there is almost impossible.