First, metabolic rate is not constant. If you feed your body few calories, your metabolism adjusts so as to use few calories. Fat cells that are stimulated to grow scream loud for their share of the energy, ensuring that they get their needs met even if the rest of you is weak, tired, and resorting to stealing energy from muscle and other tissues. This is not a zero sum game. Certain conditions may cause someone to retain water, which can add to weight independent of calorie consumption and use.
As for examples of obesity occurring concurrently with low caloric intake, take the Pima native Americans in 1905, where most of the obesity is in women, who were quite physically active, basically treated like beasts of burden. You can find the same trend in 1928 with Sioux on a South Dakota Crow Creek Reservation--very high levels of obesity in coincidence with extreme poverty. In the early 1960s, MIT nutritionists calculated that Trinidadians were getting no more than 2000 calories per day, yet they were seeing an extreme obesity problem among the females.
As for examples of obesity occurring concurrently with low caloric intake, take the Pima native Americans in 1905, where most of the obesity is in women, who were quite physically active, basically treated like beasts of burden. You can find the same trend in 1928 with Sioux on a South Dakota Crow Creek Reservation--very high levels of obesity in coincidence with extreme poverty. In the early 1960s, MIT nutritionists calculated that Trinidadians were getting no more than 2000 calories per day, yet they were seeing an extreme obesity problem among the females.