Private secretary Millicent Barnes is taking a bus to Buffalo for a new job and is caught in the storm at a bus depot. When she notices that the bus to Courtland is running late, she goes over to the ticket agent and asks when he's expecting it. He irritably says that he doesn't know and tells her to stop asking him every fifteen minutes. Millicent says that it's the first time that she's asked him since she arrived, but he says that she asked him fifteen minutes ago and fifteen minutes before that. As Millicent suggests he could be a bit polite, she notices a suitcase in the baggage area behind the desk: her suitcase. She turns around and discovers that her suitcase is still next to the bench where she was seated.
Millicent sits down and confirms that there are two suitcases. She asks the ticket agent about the second suitcase and he says that she checked it in. When Millicent turns to show him her bag, she's surprised to discover that it has vanished. The ticket agent tells her to stop bothering him and goes back to his magazine. Millicent goes to the ladies room and washes up, and the attendant asks if she's okay. When the secretary wonders why she's asking, the attendant explains that Millicent came in a few minutes ago and then walked out. Frustrated, Millicent accuses the attendant of being insane and opens the door. As she opens it, she looks back at the attendant... and sees a reflection of the waiting room, and herself seated at the bench.
Frightened, Millicent slams the door shut and the attendant offers to help. Millicent refuses her assistance and says that she's just tired. When she goes out, her duplicate on the bench is no longer there. The suitcase is back and Millicent hesitantly sits down on the bench. She considers talking to the ticket agent again but he glares at her and Millicent talks to an elderly couple. The woman wasn't paying attention and the man had dozed off, and neither one of them have any answers for her. Millicent looks into the baggage area and discovers that now the suitcase there has vanished.
A man enters the depot and notices that Millicent has dropped her handbag. He hands it to her and sits down, introducing himself as Paul Grinstead. Paul explains that he missed his plane and that the cab he took skidded off the nearby road. Millicent says that something is the matter and Paul convinces her to tell him. She talks about seeing things and how people are saying that she's been there before doing things that she's never done. Paul suggests that she's delirious and Millicent admits that would make sense except that she isn't running a fever. Her delusions also wouldn't make other people see her doing things she doesn't remember. Paul is at a loss and suggests that there may be someone who resembles her or is playing a prank on her.
As they talk, the Courtland bus arrives and Millicent thanks Paul for listening. As they go out to the bus and present their tickets to the driver, Millicent looks up through the doors, sees something, and runs back inside. Paul runs after her and fails to see Millicent... sitting on the bus, smirking.
Millicent faints from the shock and Paul and the attendant tend to her. The ticket agent tells Paul that the next bus won't arrive until morning but he agrees to wait with Millicent. The bus pulls out and the ticket agents turns off some of the lights, while the attendant wishes Paul well and leaves for the night. Millicent wakes up and Paul assures her that it's no bother for him to stay with her. She tells Paul that she once read a science fiction story about parallel worlds. According to the story, when someone from another dimension crosses over, they have to find their counterpart and take over their lives sot hat they can survive. Paul dismisses it as a metaphysics but Millicent is convinced that is what has happened. As she starts to panic, Paul suggests that he call a nearby friend and have him take them to the next bus stop, but Millicent ignores him.
Paul goes over to the ticket agent, who has been listening in and thinks Millicent is nuts. When pressed, Paul admits that he doesn't have a friend. He plans to call the police and have Millicent placed under observation, and hoped he could trick her into going. As he dials the police, Millicent goes to the ladies room and calls out to her counterpart, demanding that she show herself. Paul calls to her and Millicent comes out, saying that she's all right. He invites her out for a walk to get some fresh air, but a police car pulls up and the officers take her away.
Once Millicent is gone, Paul goes inside and tells the ticket agent that the police are taking her to the hospital for observation. The agent suggests that Paul get some sleep and he puts down his case and goes to the fountain. When he turns around, someone is running out the door with his case. Paul runs after the thief and discovers that he's chasing... himself. His counterpart, smirking, runs away into the night, leaving a confused Paul behind.