Strike One - S3-E5
Revealing mistake: In the kitchen, when Doug tells Carrie he knows the strike will end in 1 or 2 weeks tops because he's a Union Member, the blue shelves/cabinet on the wall behind Doug are positioned properly. Then, when Carrie asks him what the strike issues are, watch as the blue shelves move sideways behind Doug (this wall with blue shelves/cabinet is hinged to provide camera access). (00:13:00)
Strike One - S3-E5
Continuity mistake: When Doug has a look at the bumper sticker Spence has, there is one fold in it, but when it cuts there is now three. (00:14:40)
Strike One - S3-E5
Other mistake: When Carrie yells at Doug for ruining the bathroom, she yells when she is at the top of the staircase, and we can still see her legs. There is no way she could see the problems in the bathtub from there. (00:15:20)
Strike One - S3-E5
Continuity mistake: When Carrie says about smelling like the F train she has her hands together in front of her chest. When the camera changes angle, her hands are spread. (00:16:20)
Strike One - S3-E5
Visible crew/equipment: After Carrie discovers she can't take a shower because Doug ripped out the shower tiles, when she walks out of the bathroom keep your eye on the window at the right. Just as Doug lifts the hammer a crew member is visible in the bathroom's window (it's not Doug's reflection). (00:17:15)
Strike One - S3-E5
Visible crew/equipment: When Spence's taxi pulls away, a crew member, light screen and stage light can be seen reflected on the window of the taxi. (00:18:10)
Strike Too - S3-E6
Continuity mistake: At the start when Doug is having his breakfast something appears on top of his pancakes. (00:00:30)
Strike Too - S3-E6
Continuity mistake: When Carrie puts her bag on at the start it swaps arms instantly. (00:02:30)
Strike Too - S3-E6
Continuity mistake: The writing on the blackboard changes between shots when Doug writes his name. Note the R in Mr. (00:15:10)
Strike Too - S3-E6
Continuity mistake: When Stephanie is placing cones, she is looking at Doug with a concerned look upon her face. When the camera changes she's now facing away and laughing. (00:16:20)
Strike Too - S3-E6
Continuity mistake: At the end, when Carrie talks about going upstairs, she has her hand on Doug's arm on the back of the sofa. When the shot changes, it's now lying in front of her. (00:20:20)
Strike Out - S3-E7
Continuity mistake: The bottle of Lipton Ice Tea on the table rotates on its own without anyone touching it. (00:03:20)
Strike Out - S3-E7
Continuity mistake: When Arthur sits down for his meal, there is a bottle on the table. The lid is either on or off depending on the shot. (00:03:30)
Strike Out - S3-E7
Continuity mistake: When Carrie and Kelly are setting the table, Kelly is handing plates to Carrie. She hands Carrie a plate and has one left in her hands. The camera cuts, and she now has two plates in her hand. (00:06:20)
Strike Out - S3-E7
Continuity mistake: After Doug says "Let's hit the streets", Doug, Arthur, Deacon and Kirby go on a mini crime-spree. During this montage each characters outfit completely changes several times. They are also dressed very lightly for late October/early November in New York. (00:13:00 - 00:14:00)
Visible crew/equipment: During the flashback, while Doug is waiting on the porch Carrie rushes out then gets into her car to take off, and in the next shot facing the front door, the reflections of the camera operator, wearing a white baseball cap, and the camera are visible on the window. (00:12:35)
Visible crew/equipment: During the flashback, when Carrie walks over to Doug at the side of the house, and he zips himself up, in shots facing Doug, a crew member and production equipment are visible on the right side of the screen. (00:20:10)
Visible crew/equipment: When Doug sneaks out of Carrie's house and finds her having a cigarette by the car, a boom mic shadow can be seen on the bush beside Doug. (02:53:45)
Continuity mistake: When Carrie suggests about them looking after the kids to Doug, he has his arm up by his chest. When the camera changes it's lowered. (00:04:20)
Visible crew/equipment: When Carrie notices that Doug has walked out the room, there are some tape markers on the floor from where Doug was stood. (00:09:40)
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.