What was the nature of your chest infection? Many people present to their doctors
with symptoms of runny nose and a cough and get diagnosed with a "chest infection". In reality, there are many different types of chest infections. The two most common are pneumonia and bronchitis
. They often both will make you feel crummy and cause a cough, but only pneumonia is a bacterial cause and therefore needs antibiotics. Most likely what you had is bronchitis which most of the time is caused by a virus. A viral infection cannot be treated with antibiotics. It is so common for people to not get better after starting antibiotics for those types of symptoms simply because there is no bacterial infection to treat. If it were a bacterial infection, the antibiotics would treat both your chest infection and your sinus infection. Likely you have a viral infection that has moved into your upper respiratory tract making your sinuses congested.
To sort this out, I suggest that you schedule an appointment with your primary care physician
. He or she can perform a thorough exam of your upper respiratory tract and decide what type of additional treatment you need. Good luck.