The spots on your fern could be either a bacterial or fungal disease. It is close to impossible to give a definitive answer on diseases without proper lab testing. Is it possible that it is being kept too wet (we know, hard for a fern) and is not getting enough air circulation? You might want to seriously consider removing and properly disposing (do not compost) of affected plant parts and review your watering practices. Try watering the soil beneath the plant rather than overhead as that splashes and spreads the disease. You might also want to take a bagged sample of affected plant parts to your local independent garden center for a more definitive answer and possible suggestions for a solution based on your particular situation.