Interestingly enough, I've heard that BMI mostly under-predicts problems. There are a lot more people at a higher body-fat percentage now than when BMI was developed, meaning that BMI assumes that the average person is leaner than they are.
Back then, people in the "normal" band had more muscle and less fat than they do today. From what I understand, body-fat % is a much better metric for individual health outcomes, and BMI is a rough proxy to that. Taking two populations of the same BMI, one from say 40 years ago and one now, the one now will be fatter and less healthy.
The core problem is that BMI is a proxy for what you're actually trying to measure: body composition. Height and weight gets you there on average, if you make some assumptions about the average composition. If average composition gets worse, BMI becomes more inaccurate.
If you're a "normal" weight, but 25% bodyfat, you're not in good shape. But you may think "oh, I'm okay" rather than "I really need to adjust".
It seems obvious to me that BMI will tend to error by understating the problem. That means that if average BMI is increasing, the problem is getting worse much more rapidly than you might think.