The study referenced in the BBC article can be found here http://www.thelancet.com/journals/la...033-9/abstract
and, as Feever pointed out, it was derived from an important study that looks at more than 1M people over many decades (Clinical Practice Research Datalink, here
). The author's interpretation of the data
Being underweight in middle age and old age carries an increased risk of dementia over two decades. Our results contradict the hypothesis that obesity in middle age could increase the risk of dementia in old age. The reasons for and public health consequences of these findings need further investigation.
They are not saying overweight is better, but they are saying underweight beginning in middle age may be worse. There are other risks to being underweight in old age we already know, such as malnourishment and accidents resulting from frailty.