Only your doctor can determine the type of infection your child has and whether an antibiotic is necessary. It is natural for your child to begin feeling better soon after starting antibiotics because they've begun their job of knocking out harmful bacteria. But just because your child's symptoms may be gone, does not mean that the antibiotics have finished their work. If you stop antibiotics before the full course is finished, the infection may not be completely treated. Be sure to give your child the recommended medication as directed by your child's doctor.
Kids make about half a quart of mucus from their sinuses each day.
Not every runny nose is a reason to worry. More than half of runny noses are related to viruses or to allergies. Most viral infections last 7-10 days. If your child were to visit a doctor on the second day of a runny nose and be prescribed antibiotics for a usual 7 day course - it would seem that the antibiotics were working but in fact, the virus would be finishing its cycle. Most runny noses need nothing more than time to go away.