I disagree somewhat.
My dad developed a persistent cough when he was in an extended care facility last fall. He was there primarily for rehab after having suffered a stroke-like TIA (trans ischemic attack). The care facility gave him an x-ray that came back negative, so they declared it to be allergies. We kids didn't really believe it. He'd never had allergies.
Over a period of weeks, he was worsening - particularly his mental state and diarrhea but also the cough - so we checked him out of the care facility and took him to emergency. They, too, came up with nothing and we returned him to the care facility. The cough worsened and he was losing weight. The care facility said he was no longer progressing in his rehabilitation and that it was time to take him back home. We did, but we also made him an appointment to see his own doctor. We told the doctor about the cough and our ongoing concern that something really was wrong. The doctor ordered an x-ray and told us in an hour that he was seeing something that COULD be a slight case of pneumonia. Because of my father's advanced age, the doctor prescribed antibiotics.
The cough was gone in about three days.
So, not every case of pneumonia is the same. My dad's was only SLOWLY worsening, and it did quickly vanish once he was treated.
I guess I should also point out that pneumonia can be either viral or bacterial. Antibiotics won't do a thing for viral pneumonia. If the sputum lacks color, it probably isn't bacterial. Yellow and green suggest a need for antibiotics. Brown is the worst as it may be caused by the presence of blood. If you're coughing up brown phlegm, get to a doctor.
And don't be too quick to believe 'em if they tell you you're okay.