Can my asthma make my immune system weak?

Not really: Asthma involves excess inflammation that irritates the airways, producing mucus and bronchospasm. More on the lines of an "over active" immune system. Well-done studies show antibiotics do not help asthma flare ups. BUT: mucus plugging of the airways can lead to atelectasis, and, in theory, pneumonia, though that is the exception, not the rule, unless other factors present.