My short answer is: Yes, a person who lives with bipolar disorder can certainly be truly happy. But I think I can understand the concerns behind the question.
If you have experienced disruptive or dangerous episodes of mania, you may worry that feeling happy is just a first step toward another damaging episode. Manic episodes are generally not happy or pleasant times, but some people do experience elevated mood or an exaggerated sense of well-being as part of mania. There are, however, some important differences between mania and feeling truly happy. Those key features of mania or hypomania include rapid or disorganized thinking, impulsive or risky behavior, and irritability or intolerance of any disagreement or inconvenience. Learning to live with bipolar disorder means recognizing your personal warning signs of mania or hypomania. Rather than focusing on feeling happy as a warning signs, you’ll want to identify your personal indicators of rapid thinking, impulsivity, or irritability. Those indicators or warning signs might include things you notice inside of yourself (like thoughts jumping rapidly from one topic to another) or changes in your behavior (like using your phone for calling or texting in the middle of the night).
If your family and friends have seen you go through disruptive or frightening manic episodes, they may also start to worry when you are just feeling happy. And it can certainly be irritating if you are feeling happy and a friend asks “Have you been taking your medication lately?”. People who know you well may be able to recognize some of your indicators or early warning signs of mania. Their input can be helpful, if it’s communicated in the right way. If there are people you trust to give you that feedback, it’s best to talk at a time when you are not feeling manic or speeded up. Ask them about the warning signs they think they have noticed in the past. If you agree that those warning signs are valid, then talk about the best way a friend or family member might communicate concern—including specific helpful and unhelpful things to say.
Some people find that medications used to treat bipolar disorder make them feel dull or flat—and less likely to experience happiness or positive feelings. That is not the goal of medication treatment for bipolar disorder, it’s an unwanted side effect. If you notice that your medication has that side effect, be sure to ask your doctor if there are better alternatives for you.