The studies @Westside PUFAs posted had follow-up periods of 21 and 27 years. Health evaluation were done when the participants were in their 40s.
So, it was truly obesity in middle age then that was detrimental. The study I posted was on elderly. What I am not clear is if the study looked at whether people got fat when they reached "old age" or were obese before that. If it is the former, than as counterintuitive it sounds, the statement "getting fat when you reach senior citizen age may protect you from dementia" would be true according to that study.