These are useful pages to explain the role of insulin in the Lipid metabolism:
http://www.vivo.colostate.edu/hbooks/pathphys/endocrine/pancreas/insulin_phys.html
http://www.metaboliceffect.com/science-insulin/
Essentially, aside from its role in the carbohydrate metabolism, insulin also has a prominent role in the lipid metabolism, causing fat storage and inhibiting fat burning. If you are insulin resistant and therefore producing too much insulin, two things happen. You store more fat and use it less as an energy source. You also generate more glucose from the liver due to the insulin resistance, and guess what, you store it as fat as your muscles don't use it as fuel fully.
If you are injecting insulin, you are doing the same thing. The insulin stimulates the storage of fat. Insulin allows glucose into the muscles and triggers the liver to store glycogen and as soon as your glycogen stores are full and you don't need the glucose as fuel, you store it as fat.
Ergo, yes, taking too much insulin and eating too many carbs, i.e. not balancing what you consume effectively, can cause you to gain weight as fat as a type 1 diabetic.
If you are eating a lower carb diet and taking your energy resources more from non-carb sources, less insulin is required and there is therefore less conversion of glucose to fat, less signalling to fat cells to create fats and less inhibition of the ability to use fat as an energy source.
They don't teach you that on DAFNE.