Unlikely. Allergy causes inflammation around the opening to the sinus which may cause an infection. If you aren't allergic, that is unlikely.
Sinusitis. Sinusitis in the most simple of terms means inflammation of the sinuses. Cat dander can be very irritating and even though you are not technically allergic to the cat, if the cat goes outdoors it can pick up pollens that stick to its fur and/or it could also be the quantity of cat dander you are exposed to (sleeps with you) that can cause symptoms.