Can cystic fibrosis cause chronic bronchitis?

Not the same: Cystic fibrosis (CF) is a genetic disease that may be characterized by a chronic inflammation of the brochi (chronic bronchitis), but it is not the same as the entity "chronic bronchitis", as commonly used, that is often cause by smoking. If the term "chronic bronchitis" is taken literally, then yes, CF can cause it, i.e. Cause a chronic inflammation of the brochi (parts of the airway).

Bronchiectasis: Chronic airway infection in CF is common; it isn't the same as "chronic bronchitis" precisely but has similar traits (copious purulent sputum production, discomfort, airway obstruction). More commonly the chronic infection and mucus plugging lead to airway destruction, dilation in a condition known as bronchiectasis. Google cts of the chest for both and take a look at the differences.