Tha adaptive immune system is highly specific, but the innate immune system plays an important role in the start of an infection. When it is busy trying to kill one thing you are at slightly higher risk of a different infection- this is largely why secondary infections are a thing.

The adaptive immune system kicks in a bit later, and is certainly more powerful, but a functional innate immune system should prevent a lot of pathogens from causing infections before the adaptive response has really had to worry about it, by phagocytosis etc. Especially if the exposure dose of virus/pathogen is fairly low.