Whenever cure/remove disease is cast, the target must make a CON-save.
If failed, the disease became (or was already) resistant to this particular cleric/normal person's version of cure/remove disease. The diseased individual still suffers from the disease, and may only be cured by another person but this time with a penalty (+1/-1 or similar). The penalty accumulates.
(Antibiotic resistance is serious problem, people!)