Although your risk of developing diabetes is increased if you weight more, anyone at any weight can develop type 2 diabetes if they eat enough carbohydrates.
The reason for this is that type 2 diabetes (the most common form, often referred to as adult onset diabetes) is the consequence of 2 major factors, carbohydrate intake and genetics.
Diabetes is driven by an underlying medical condition called insulin resistance. Insulin resistance also leads to high blood pressure, kidney disease, elevated triglycerides, and the blood vessel damage that causes heart attacks and most strokes.
Excessive carbohydrates trigger insulin resistance and can lead to diabetes. Many people will also become overweight or obese if they eat too many carbohydrates. This is a separate metabolic effect of carbohydrates.
Others will develop diabetes but will retain their normal weight. This is because they are only susceptible to the insulin resistance effect and not the obesity effect of carbohydrates.
The likelihood of developing diabetes with or without obesity is a factor of the genetic material that you inherited from your parents. A family history of obesity and diabetes implies that you have a genetic susceptibility to the harmful metabolic and weight-gaining effects of carbohydrates.