The cat in "Once upon a Time" plays a significant symbolic role. First, the cat symbolizes bad fortune. Black cats are associated with bad luck, and since Gordimer also includes a "witch" in the story, the connection becomes even stronger since cats and witches are paired in folklore.
Second, the cat symbolizes the unconquerable fear the family has for most of the story. The cat is able to get into the house despite the bars and to get into the yard despite the high wall. This shows the family's fears continue to beset them despite their efforts to protect themselves.
Third, the cat represents false alarm. The cat—as well as other cats in the neighborhood—routinely sets off the burglar alarms in people's homes. This suggest the fears the suburban residents have are "false alarms." The alarm they experience is not grounded in reality. In the same way that the boy who cried wolf was not believed when the wolf finally did attack the flock, so too do the continual false alarms set off by the cats and mice prevent the residents from discovering the real burglaries that occur. By overreacting to the threats, the suburban residents are unable to deal effectively with their true problem.
Finally, when the Dragon's Teeth fencing is installed, the cat stays inside the enclosure. The husband calms his wife's fears for the cat's safety by stating that "cats always look before they leap." In this way, the cat represents wise foresight. At the beginning of the story, the family has used wise foresight in fencing off the swimming pool to protect their son and his friends. Now they have become so obsessed with keeping out the undesirable elements that their cat has become wiser than they are. They should be able to foresee the danger their fence causes to their child—not just physically, but even emotionally and socially—but their fears blind them.
The cat is symbolic in multiple ways in the story, representing ill fortune, unconquerable fear, false alarm, and wise foresight.