No, it is not possible for a good to be both normal and inferior. These are two categories that are opposites of one another so it is completely impossible to be both at once. Let us see why this is so.
The definition of an inferior good is that changes in the demand for this good are inversely related to changes in consumers’ incomes. That is, when the consumers’ incomes rise, demand for these goods falls and when consumers’ incomes fall, demand for these goods rises. We can imagine that something like instant ramen noodles would fall in this category, with people buying more of them as their incomes fell. If their incomes rose, they would tend to want better things and demand would fall.
The definition of a normal good is that changes in demand are directly related to changes in consumers’ incomes. That is, when consumers’ incomes rise, so does demand for the product. Most products are like this. The more money we have, the more of a product we can afford and therefore the more we buy.
As you can see, these two definitions are mutually exclusive. If consumers’ income goes up, demand for a good cannot go up (normal good) and down (inferior good) at the same time. Therefore, a good cannot be both normal and inferior.