I just got my ACT score and I got a 21. I want higher. Is it unrealistic to think I can raise my score by 5 points? How can I raise my score?

You can definitely raise your score. Can you remember what your weaknesses were? Write down what you had trouble on and go back and study that until you can "breathe" it. Take more practice tests. Raise your weaknesses in a certain area to a level of understanding.

This is only the beginning of success in wisdom. Practice makes perfect. In this case, practice tests make perfect.