OK we need to know the full story here to help.
Yes it is possible for antibiotics to fail to remove chlamydia.
Transmission rates are very high for chlamydia. It would be rare to have unprotected sex with someone that had a genital infection more than a dozen times say and not be infected. Not impossible, but very implausible.
Ok, so I'm not a doctor but my situation seems similar to yours. About a month ago, I tested positive for chlamydia. I took the 2 gram dose of Azithromycin all in one go. My symptoms disappeared for about 3 weeks or so and then suddenly came back one morning. It was the strangest thing because between the time I took the medicine and 3 weeks later, I didn't have any form of sex with anyone. I have read that chlamydia can live in the gut and come back to reinfect someone who has been treated for it, which might be my situation. So, I would say that the answer is YES, azithromycin and fail because it failed in my case. I'm living proof.