Continuity mistake: Ritchie asks if Doug is trying to get him drunk and has both hands out of his pockets. Camera cuts and one hand is now in his pocket. (00:05:00)
Continuity mistake: When watching the film, Doug's legs goes from the table to the floor between shots. (00:03:10)
Train Wreck - S1-E20
Continuity mistake: When Carrie turns up at IPS, she holds out the coloured char with two hands then one depending on the shot. (00:09:50)
Train Wreck - S1-E20
Continuity mistake: When Ritchie is pretending to be Carrie, Doug has hold of his cue with both hands. Camera cuts and he's not holding the tip any more. (00:06:05)
Continuity mistake: When driving the Mini, the cars around them are constantly changing. (00:12:05)
Continuity mistake: When Carrie asks Doug what he's been doing, her hands change place instantly. (00:07:20)
Continuity mistake: Just after picking the big turkey, Carrie's bag keeps changing place in the trolley. Note its position in relation to the handle. (00:06:00)
Continuity mistake: When the IPS guys are discussing the Superbowl party, one of them takes a drink. He then lowers his can. When the angle changes, the can has changed hands. (00:04:20)
Continuity mistake: When Doug is handing the deodorant back to Arthur the way he holds it changes between shots. (00:06:15)
Continuity mistake: When Tim asks Doug if he wants to tee off back at the start of the hole rather than halfway down the fairway, he is pointing with his golf club. Camera cuts and he isn't. (00:15:35)
Soft Touch - S2-E22
Continuity mistake: When Doug is getting into bed while talking to Carrie about selling water filters, his grip on the duvet changes instantly. (00:10:00)
Continuity mistake: When they are in the kitchen Carrie's purse switches from shoulder to shoulder when she tells Doug the good news.
Continuity mistake: When the nurse walks away from Doug, she walks away, camera cuts and she starts to walk away again. (00:07:25)
Deconstructing Carrie - S7-E15
Continuity mistake: Doug is sitting on the couch making club music sounds and moving his hands when Carrie reaches out and touches his face to make him stop; then his left hand is instantly on the arm rest. (00:13:10)
Party Favor - S2-E13
Continuity mistake: When Doug is reading his speech to Arthur, Arthur's legs are crossed or uncrossed depending on the angle. (00:09:30)
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)
Continuity mistake: When Carrie asks Arthur how a legless dummy will protect her on the subway, her arms are either on the table or off, depending on the shot. (00:10:30)
Soft Touch - S2-E22
Continuity mistake: When Doug is telling Arthur about the filters, there is a leaflet on the table. Between camera cuts, this leaflet is constantly changing position. (00:13:25)
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.