On stardate 41697.9, the USS Enterprise is on course to Sarona VIII for shore leave. Captain Picard has begun his recreation in the ship's gymnasium, practicing fencing with Lt. Dean. He and Dean are sparring against each other when suddenly, there is a shift in time and the last few seconds repeat. Picard calls the bridge and Commander Riker reports that the ship has passed through a distortion in time. Picard goes to the bridge where Lt. Worf receives an automated distress call from Dr. Paul Manheim, stating an urgent need of assistance. Riker recalls the story that Manheim was ejected from the Federation Science Institute for conducting unauthorized experiments. Picard orders an intercept course.
Picard goes to the holodeck and brings up a program for the patio of the "CafÃ© Des Artistes" where he takes in the 24th century Paris skyline. He observes a lovely young girl at a nearby table talking to a friend and overhears her conversation about her boyfriend's decision to join Starfleet which would result in his leaving her behind on Earth. It bears an uncanny resemblance to Picard's past, when he made a similar decision against the wishes of a former girlfriend, and the memory pains him.
Picard returns to the bridge where Riker reports the time ripple was also felt by the USS Lalo and the colony on Coltar IV. La Forge then picks up new coordinates relayed from the original source of the distress call, but it leads to the isolated Vandor system. Apparently, Manheim doesn't want to be found. Heading to Vandor, they find a battered planetoid orbiting a red giant and a pulsar. Sensors detect an impenetrable force field surrounding the planetoid which is emanating from a facility on the surface. They then receive a hysterical call from a woman who says her husband, Dr. Manheim, is injured.
The two are beamed directly to sickbay where Dr. Crusher reports Manheim is fading in and out of consciousness. Picard arrives to see Manheim's wife Jenice standing in the doorway. She is surprised at the sight of him and calls him "Jean-Luc". He takes her into Crusher's office where she says her husband has been working in the lab for days, but she doesn't know what he's been doing. Picard says he will beam an away team down to investigate, but Jenice warns of her husband's elaborate security systems.
Picard walks to the bridge with Data and Riker. When the three enter the turbolift, they see themselves for a moment standing behind them. Data indicates the "Manheim Effect" is becoming more prevalent. On the bridge, Data reports the presence of an energy beam in the main lab of the facility, suggesting it may be keeping the dimensional door open. He states Manheim made pioneering efforts in the theories of linear time before leaving the institute and Picard wonders if any of it has to do with whatever is going on down there.
Riker, Data and Worf beam down, but when they materialize in a corridor of the station, their forms become erratic and they bounce back to the Enterprise.
Crusher calls Picard to let him know that Dr. Manheim has finally woken; the captain and Data rush down to talk to him. Manheim is confused about what happened, incoherently saying he has been to the "other side". Picard tells him that whatever he has done is affecting the entire galaxy. Manheim says he was experimenting with relationships of time and gravity and states he was able to produce a funnel into another dimension. Manheim says he doesn't remember his security codes, but provides transport coordinates that he thinks should be safe. He then falls back to unconsciousness.
Picard heads to the observation lounge where Jenice enters looking for him. She asks him why he stood her up in Paris. He admits it was fear in losing her and himself. She admits she still loves him. Picard is fearful that if the time experiment goes out of control, he'll lose her again. Down in sickbay, Manheim awakens. Picard arrives and Manheim tells him that he deserves Jenice; he knows she has always loved him. He wants to be sure that whatever happens to him, Picard must promise to take care of her.
Later, Data reports that they have bypassed the security system and the captain gives him approval to beam down alone. Data beams down into a corridor and encounters blasts from two automated security phasers. He fires at the weapons to disable them. Inside the lab, he finds a column of energy which makes waves of visual distortion. He calls to the Enterprise and indicates that he can add antimatter to the dimensional matrix and restabilize the energy flow. This should cause the dimensional window to collapse and end the "Manheim Effect".
Data removes the antimatter container and readies to insert it into the dimensional column. He looks back and is amazed to see his past self entering the room. A third Data appears and repeats the actions of the second. Data realizes that only one of his counterparts is in the correct timeline. They decide to work together simultaneously; as one, the Datas insert the antimatter, causing the column of energy to flare up before finally vanishing. The hole has been plugged and Data's doubles disappear.
Aboard the Enterprise, Manheim makes an instant recovery. Jenice is ecstatic. He assures her that everything is okay now. Manheim thanks Picard for his help and assures him he will not conduct such a foolish experiment again.
With everything settled, Picard heads back to the holodeck cafe program. He sits at his table and smiles when he sees Jenice, who came to say goodbye, enter. They share a glass of wine together. Jenice and her husband head back to their facility as Picard goes to the bridge. He tells Riker he remembers them being on their way to Sarona VIII, mentioning a great bar there, but he forgot the name. Deanna recalls the place, the "Blue Parrot CafÃ©", and Picard tells her she's buying.