When you are battling an infection is your immune system lower? are you more likely to get sick?

Depends on the infection. "Infection" is a broad term that can be used to describe any kind of illness your body is fighting. Some viruses and conditions do in fact impair your body's ability to combat them and anything else, meaning your immune system has been weakened.

If you have a weak immune system in general, yes, you are at a much greater risk of getting sick and falling into a bad state while ill.