Warning: spoilers from Episode 3 and the original novel. Early on in the third episode of The Handmaid's Tale, the other members of Offred's household began to suspect that she was pregnant. She showed no signs and her period was only a few days late, but everyone was so attuned to any signals that indicate pregnancy that they immediately jumped on the possibility. And it produced an immediate change in Offred's treatment: she was given a nicer lunch with a little dessert, and the normally icy Serena Joy showed her some warmth. But does Offred get pregnant on The Handmaid's Tale? It seems much too soon to tell.
Pregnancy would give Offred a certain amount of power and special treatment. So few women are able to conceive in Gilead, and they are even more unlikely to give birth to a healthy baby. When it was suspected that Offred was pregnant, she was even spared violence at the hands of the government for her failure to report on her friend Ofglen. However, towards the end of Episode 3, Offred got her period, making it clear that she was not pregnant after all. As soon as she revealed that information to Serena Joy, the change was obvious: in mere seconds, Serena Joy went from loving and kind to throwing Offred around and screaming in her face. Pregnancy was the only currency Offred had anymore, but it was tenuous and could be lost in an instant.
Serena Joy was so ready to jump on Offred's pregnancy with such little evidence that she got her hopes up over nothing. It was almost ridiculous, because there were no pregnancy tests or trips to the doctor to find out for sure; the only reason they suspected anything was because Offred's period was a few days late. Then when she was not pregnant, all the blame was on Offred once again even though she hadn't been the one to make the assumption or the announcement in the first place.
In the book, Serena Joy's desperation for a child pushed her to extreme measures. She arranged for Offred to sleep with Nick, the Commander's chauffeur (and possible government spy) to increase her chances of conceiving. In exchange, she promised Offred a picture of the daughter who was taken from her before she was forced to become a handmaid. Offred ended up finding some solace in her relationship with Nick, and eventually did come to believe that she had gotten pregnant.
If that storyline does happen on the Hulu series, it probably won't happen any time soon. Everyone might be waiting for Offred to get pregnant, but it hasn't happened yet.