When I was a child, I had this. A 6 week cycle of cold -> sinus infection which required antibiotics to normalize, 5 weeks of normalcy, then another one. I didn't grow at all during 1st grade due to constantly being sick.

I had bad adenoids - they're like tonsils but in the back of your nose. They are supposed to shrink away to nothing as you grow but mine were hypertrophic and blocking part of the drainage and thus causing constant infections. I had them removed in a quick and easy surgery in second grade, the cycle ended instantly, and I caught up with my classmates quite quickly.