It is good for predicting in general populations, but can easily fall apart with individuals. It doesn't take into account what the makeup of your mass is. Muscle is denser than fat, so a muscular man and a fat man of the same height and weight would have the same BMI, but vastly different health profiles.
While it can "easily fall apart with individuals", the difficulty lies in finding an individual to measure that will make BMI fall apart.
And that's because the huge majority of us is of the kind of complexion and body type where BMI works well.
So, sure it can give false positives, but for a quite small fraction -- besides any other rough medical heuristic does the same (e.g. signs of having the flu can also be caused by 200 other reasons).
A vastly more accurate proxy is waist circumference.
BMI was developed largely because the proxy data (height and weight) are so readily available, especially in databases. Few doctors take waist measurements, and doing so is slightly more complicated.
> the difficulty lies in finding an individual to
> measure that will make BMI fall apart
I was 212lbs this morning, giving me a BMI of 29.6, which has me at the border of "Overweight" and "Obese Class I (Moderately obese)", and is high risk.
I don't think you'd think I was a weird shape if you looked at me.
Both measures put you in an at risk categories. Why doesn't that concern you?
How many runners do you know who would complain about BMI? Or weight lifters? It's the fat people who don't like it.
Calipers are also $10 and measure body fat accurately enough for health purposes (body building may be an exception in this as well, but only in extreme cases).