Is it possible to treat chronic, nonbacterial prostatitis?

Possible. It is certainly possible to treat chronic abacterial prostatitis but is very difficult. Treatment at this point is usually for comfort measures to include anti-inflmmatory medications, sitz bath, and avoidance of certian foods and behaviours that provoke symptoms. The treatment may not cure the problem but will help control the symptoms to make the patient more functional on a day to day basis.
Yes. If prostatic secretions after prostate massage shows evidence of inflammation but no bacteria, often a short course of antibiotics is tried nevertheless. Other treatments include periodic prostate massages, instructions to ejaculate every 3 days, medication to relax the bladder outlet,anti-inflammatory medications, pelvic muscle exercises. A urologist is best qualified to treat this difficult dis.