Bronchitis is contagious during the period that the infection is still active.
More Info: Acute bronchitis caused by a virus is the most contagious during the first few days, if a fever is present, and as long as a cough persists. 
Some Causes of Bronchitis Are Contagious, Some Are Not
Bronchitis is a swelling in the bronchial tubes that leads to mucus collecting in the lungs and has a variety of causes. Bronchitis that is contagious is caused by either a virus or bacteria. Bronchitis can also be caused by smoking, pollutants, or irritants. This type of bronchitis is not contagious. 
You Can't Actually "Catch" Bronchitis
Up to ninety-five percent of bronchitis cases are caused by viral infections, most frequently those associated with the cold and flu.  Acute bronchitis is a complication that can arise from these viruses if they move into the bronchial tubes.  Therefore, it is not the bronchitis that is actually contagious, but the virus that causes the bronchitis.
How Bronchitis Spreads
Just like any illness caused by a virus, those that cause bronchitis can be spread through contact or through germs that travel through the air. You can contract the virus by spending time with a person who is ill, especially if that person is running a fever. The virus that caused bronchitis in the infected individual can be spread through coughs, sneezing, or contact with an object that has been handled by someone who is suffering from the illness. The best defense against spreading bronchitis is to wash your hands thoroughly and cover all coughs with a tissue. 
While Running a Fever
Fever may accompany an illness such as bronchitis and is a sign that the body is actively attempting to fight off the virus. If fever is a symptom of your bronchitis, you should stay at home until the fever breaks. Confining yourself to bed during this highly contagious stage of bronchitis will also allow you to rest and recover more quickly.  It is best to wait at least 24 hours after your fever breaks before you return to regular activities. 
As Long as Coughing Persists
Because most viruses are spread through aerosols, anyone suffering from bronchitis where the infection is still active is still contagious as long as they are coughing or sneezing. According to the Centers for Disease Control, aerosols can spread up to six feet away to infect another person.  Make sure that you take precautions to protect other people from your coughs and sneezes so that you do not spread the illness among your friends and colleagues. Wash your hands often and do not share items such as drinking cups or silverware until your cough has completely gone away. Depending on the virus, it can continue to survive on contaminated surfaces for several hours, so it is important to clean your hands often.
 WebMD, Jennifer Shu, MD
When Is Bronchitis no Longer Contagious? Baby's First Year Community - Support Group.
Is Bronchitis Contagious?