Can a sti cause you to not get pregnant even if you've been cured?

STDs: There are several possibilities, but as an example if you had infection with gonorrhea or Chlamydia or other atypicals they may cause scarring of the fallopian tubes which lead from the ovaries to the uterus and impede progression of the ova downwards. If fertilized they may result in ectopic pregnancy. There are other issues, too detailed to discuss here, but see an infertility specialist.

Yes, sometimes: Chlamydia and gonorrhea in women can get into the fallopian tubes. Sometmies this scars the tubes, blocking them. When that happens, sperm and egg are blocked from contacting one another. This is called tubal infertility. Treatment will cure the infection, but the tubes remain blocked. Sometimes this happens with silent infection, i.e. no symptoms, especially for chlamydia.