Is it normal for someone to have trouble breathing when it is cold outside?

Depends. Breathing problems from cold can be cold induced bronchospasm or asthma. Get to the doctor, get tested or at lest checked out and treated.
Not necessarily... Some people with asthma or COPD may feel short of breath when breathing cold air. This symptom should be evaluated by a physician as there may be treatments that could be helpful. Meantime, covering the nose and mouth when exposed to cold air may also help.