Can a diabetic build up resistance to insulin?

(a careful) yes. Type 2 diabetics have Insulin resustance, but early on are able to make extra Insulin to overcome the resistance. Over time, they are less and less able to do that. Then they need medicine, and eventually need to take insulin. If a type 1 diabetic gains weight, he can also develop Insulin resistance and his doses will need to be increased. Weight loss decreases resistance in both types 1 & 2.
Insulin resistence. It is possible to develop Insulin resistance with time. This usually happens in type 2 dm, and it is the Insulin receptor that develops the resistance. Many time it is overcome by increasing the dose of Insulin and at other some medications may help. Obesity increases the risk in fact metabolic syndrome is a specific syndrome with high risks and Insulin resistance.