Yes, this can and does occur, in the case of both types of diabetes. However, in the case of type 1 diabetes, the disappearance is very predictably temporary and the diabetes will almost inevitably return within months or a year or two. The reasons for it are complex and relate to the fact that type 1 diabetes is often diagnosed under conditions of physical stress.
When the diabetes is treated and the stress to the body has resolved, there may be sufficient remaining insulin-producing capability in the pancreas to keep the blood sugar normal under most circumstances. Eventually, however, the pancreas fails and permanent diabetes supervenes. In the case of type 2 diabetes, although the hereditary factors causing it cannot be reversed, the environmental and lifestyle factors can be changed. The latter are the cause of the explosive rise in the number of people affected with diabetes in recent years.
Although we do not generally refer to type 2 diabetes as being cured once it has been diagnosed, it can certainly go into remission (cease to be an active medical problem) for long periods, depending on how successfully the causative factors are addressed. Because excess weight and sedentary lifestyle are the two most important and serious causative factors, weight reduction and adherence to an exercise program can often put type 2 diabetes into remission for as long as they are maintained.
Some-times, all medications, even insulin, can be stopped and all measures of diabetes, including blood glucose and hemoglobin A1c (HbA1c—the measure of your blood sugar control averaged over 3 months), will completely normalize. Other associated conditions, such as high blood pressure and cholesterol abnormalities, will also improve greatly. The possibility of stopping all medications is more likely to occur earlier in the course of diabetes and before long-term complications have developed. Unfortunately, many of us in modern society are unable to sustain weight reduction and exercise for long periods and the failure rate is high.