Yes they can.
It depends on the diabetes.
There is also something called transient diabetes which is generally brought on by steroids. This is temporary. Once the body regains control the diabetes will go away.
Type 2 diabetes is most common in cats.
Some cats (certainly not all or even most) can be diet controlled. Often that requires a low carb diet.
Other cats can go on what is referred to as a honeymoon period, in which they can go a period of time without needing insulin. This can last from days to years or even forever.
My own cat is diabetic, has been for almost 6 years. She did have a period of about 3yrs of not needing insulin. For her, it seemed like her pancreas could produce enough insulin on its own to deal with certain amounts of sugar, but if she were to eat forbidden food it would take about 3 days of very small doses (0.5unit) for her to get control again. She had sort of a "sputtering" pancreas I guess you can say.
She has needed insulin consistantly for about the last year and a half. Diabetics are very susceptible to infection and infection can reak havoc on bg numbers. In her case, she had a really bad dental infection that just sent her numbers skyrocketing to the point of no return.