From this NPR article Top 10 Reasons Why The BMI Is Bogus, the top ten are:
1. The person who dreamed up the BMI said explicitly that it could not and should not be used to indicate the level of fatness in an individual.
2. It is scientifically nonsensical.
3. It is physiologically wrong.
4. It gets the logic wrong.
5. It's bad statistics.
6. It is lying by scientific authority.
7. It suggests there are distinct categories of underweight, ideal, overweight and obese, with sharp boundaries that hinge on a decimal place.
8. It makes the more cynical members of society suspect that the medical insurance industry lobbies for the continued use of the BMI to keep their profits high.
9. Continued reliance on the BMI means doctors don't feel the need to use one of the more scientifically sound methods that are available to measure obesity levels.
10.It embarrasses the U.S.
The simple fact is that the BMI does not differentiate between fat, muscle and bone. A strong athlete would be considered overweight or obese. Mike Tyson at 5.5% body fat in peak fight condition would be considered obese.
The final line of the article offers a succinct summary:
It is embarrassing for one of the most scientifically, technologically and medicinally advanced nations in the world to base advice on how to prevent one of the leading causes of poor health and premature death (obesity) on a 200-year-old numerical hack developed by a mathematician who was not even an expert in what little was known about the human body back then.