It is better to say that when people have colds, they have difficulty in breathing than cannot breathe at all. Colds (also known as nasopharyngitis) are generally due to virus that affects our respiratory tract, particularly the upper part. Once affected by the virus, mucus starts to form thus blocking the nasal cavity which prevents us from breathing properly.
Simple blocking of mucus in the respiratory tract can lead to a much more effect such as the nasal congestion. It is the medical term for the blockage of the nasal passages of the respiratory tract of humans (and some animals) due irritation of the respiratory linings. Irritation happens when the blood vessels swell.
Colds in general are not life threatening but can be a symptoms of a much more serious infection or disease. For infants, difficulty in breathing is being monitored since their physical bodies are weaker compared to adult beings.
Breathing problems could be for many reasons. The most simple explanation would be that the viruses that enter the body disrupt the normal functioning of the body. This causes side-effects such mucus formation. When too much mucus is produced in the nasal passage, it becomes difficult for the inhalation and exhalation of air. There by causing difficulty in breathing.
The mucus also blocks the nose sometimes. Therefore it becomes difficult to inhale air.