The answer to this question involves one of the most problematic areas of medicine today, namely antibiotic resistance. In diagnosing a bacterial infection, a culture of your conjunctiva (pink of lower eye) must be done. The culture must come back positive for a bacteria WITH the antibiotics that are effective against that organism. It is possible that the bacteria in the eye is resistant to the first antibiotic used.
If this culture is negative, the cause is nor a bacteria. The redness could be due to a scratch of the cornea (clear layer over the eye surface), foreign body of the eye, local irritation by allergy/virus (these are usually in both eyes) or dryness in the eye (needs lubricant drops). Depending on the above causes a different treatment is needed.