Deliberate mistake: When Arthur enters the living room playing the harmonica, it is obvious he is not actually playing it. (01:23:10)
Visible crew/equipment: When Doug uses the ridiculous RV horn and startles Carrie awake, as she walks from the sofa to the front door, the overhead studio lights are visible at the top of the screen. (00:11:40)
Revealing mistake: After Doug successfully trades vacation days with Roger, it cuts to Doug's IPS truck stopping in traffic. The next few shots were filmed on the streets at Sony Pictures Studios lot. In the shot of the IPS truck's side-view mirror (as the RV approaches), note the signs on the building read "Keaton Building" and "Sound Department Courtyard." After the flashback, another building sign reads "Durante Building." These buildings are located at the north side of the studio in Culver City, CA. (00:03:20)
Answer: To answer the question directly. Carrie and Doug do not have a biological child during the show's main run. In season 3 or 4 finale (cant remember which) Carrie has a miscarriage. Later on in the show they find out Carrie has 1 working ovary. Thus reducing the chance of them conceiving. By the final season, they couple agree to adopt from China and go to collect their child in the show's final episode. While on the flight back from China it's revealed that Carrie is once again pregnant. However the last scene of the show is a flash forward showing that Carrie goes full term with that child and they are struggling with both kids.
Ssiscool ★
Adding on to this answer, I read somewhere that producers were afraid of the show having a "before baby" feeling and then an "after baby" feeling. Some fans feel that main characters, or the show in general, decline in quality after a major change. For example, Jesse and Rebecca on "Full House" after they had twin sons, and Daphne on "Frasier" after she married Niles.