Can you get cystitis, or something similar, from having sex?

Yes: This can happen due to occasional forcing of bacteria on skin to inside the urinary bladder.

Women certainly: Bacteria around the urinary opening are common in women prone to utis/cystitis + penile thrusts during intercourse can "massages" such bacteria up into bladder. Many bacterial strains thrive +multiply in urine thus causing utis, hence need to urinate after sex to eliminate both bacteria + "nourishing" urine. Men can get urethritis (std) from intercourse, cystitis unlikely as male urethra is longer.