Consistently training to failure (which is what you're talking about) is a relatively controversial training technique. Some people advocate it as a way to push through plateau periods while others (myself included) consider it overtraining.
I think the only really beneficial thing about training "until they broke down or had to throw up" is that it will tell you exactly how far you can go before your body gives up. This is valuable information to know if you're a serious endurance athlete or someone like Lance Armstrong.
I'm not a fan of training to failure mostly because it is a safety issue for me (that and I don't feel like it is necessary). I weight lift by myself (no spotter) and I trail run by myself (I don't want to get exhausted five miles up a valley with the sun going down).
As an aside, training to failure shouldn't be confused with high-intensity interval training which has some well documented advantages.