Can I become repeatedly infected with chlamydia?

Yes. Chlamydial infections are treated with antibiotics. However, if the person again has relations with another partner who has an active chlamydial infection, that person will become reinfected and will require retreatment.
Yes. Chlamydial infections can recur with re-exposure. Both (all) partners must be treated simultaneously to prevent reinfection. Condoms can protect against reinfection.