As @Kayak8 and others have noted, the body can defend itself against many, but not all infections. For those that the body can fight off naturally, the time for antibiotics to help speed the process probably decreases with time from onset of the illness. If you wait a few days, your body is already producing antigens and the antibiotic just helps speed recovery. Other bacteria are stealth fighters with the ability to avoid the body’s natural defenses. With these, antibiotics are the only cure, and the sooner they are begun the better. Allowing the infection to spread throughtout the body and penetrate areas hard to treat with antibiotics could be the difference between complete recovery, permanent impairment or even death. So the take home is, if you don’t know the specific nature of the bug that’s plaguing you, see the doctor as soon as possible.
Another note. While antibiotics can work wonders against bacteria, they are useless in fighting viruses. Sinusitis can result from a viral, bacterial or fungal infection, and also from allergies and autoimmunity issues. How to best treat sinusitis is entirely dependent on what’s causing the flareup.
When patients go to the doctor with a cold (viral infection) or non-microbial sinus infection and demand an antibiotic the doctor will often prescribe one. More money for her, and the placebo effect may even help the patient. But every time we do that we are allowing whatever bacteria happen to be in our body in even trace amounts at that time the possibility to evolve immunity to yet another antibiotic. That, and the overuse of antiseptic cleansers are what have lead to the emergence of MRSA and other drug resistant pathogens. Leave it up to the doctor to decide the best treatment option. Don’t self diagnose.