Is it possible to catch bronchitis?
Bronchitis is not an infection; it is an inflammation of the bronchial passages that lead to the lungs. So no, you can't catch it like you would the common cold or the flu. But someone who has chronic bronchitis is more susceptible to respiratory infections, and it could be possible that those infections could be passed along if the person were to cough on you or something along those lines.
Does that answer your question?
P.S. Here are some links that might be helpful to you as you're dealing with this health issue:
10 Facts About Treating the Common Cold