I feel bloated and hungry at the same time. Is this normal?

Unusual. Bloating is usually caused by excessive gas production or by inability of normal amounts of gas to descend into the intestines for eventual passage. Frequently people try to belch and all they accomplish is swallowing more air and adding to the bloat. When you're hungry you may also swallow air inadvertently. Alternatively your stomach may not empty normal gas when you're very hungry.
No. Not necessarily normal but common. Bloating is a very non-specific symptom associated with everything from irritable bowel to ovarian cancer. It is usually a benign symptom but you should discuss it with your doctor.