Peter Mitchell, an architect, Jack Holden, an actor, and Michael Kellam, a cartoonist, move into & remodel a Manhattan penthouse. After a large housewarming party, Jack leaves to film a movie in Turkey. Before he leaves the country, he calls his roommates and tells them that a friend of his needs a small package held at the penthouse for a few days until his associates come to pick it up.
Peter returns home from jogging one morning and finds a small basket with an infant girl in it at their door. Attached is a note from a woman named Sylvia, who writes in the note that she's leaving the baby, named Mary, with Jack, the father. Peter and Michael take the baby in, assuming it's the "package" Jack's friend meant to leave. Frantic because they don't know how to care for an infant, Peter rushes to the store for supplies and Michael entertains little Mary, who won't stop crying. When their building manager, Mrs. Hathaway, arrives at their door with a small package left for the men at the desk, Michael accepts it and throws it carelessly onto the couch. Peter comes back with diapers and food and the two spend a significant amount of time changing Mary's diaper and washing her when she soils herself and their couch. Caring for Mary quickly consumes all their free time and cancels their social engagements, though they do an admirable job.
A few days later two shady men arrive and ask for the "package." Michael and Peter assume they've come for the baby and hand Mary over to the men. Peter is suspicious and sits down on the couch, finding the small package Michael had thrown there. Peter realizes that it is the actual package the men were after and rushes downstairs to stop them before they drive off. In the main foyer of their building, Peter trips and the package bursts open, spilling its contents: small bags of heroin. He meets the two men on the street and tries to take Mary back, saying he made a mistake. When a mounted New York cop becomes involved, the men speed off and the cop stops Peter from retreating. The cop demands to see ID and goes upstairs with Peter. Peter gives Mary and the drug-filled parcel to Michael, ordering him to hide it. When the cop sees Peter's ID is legitimate, he leaves, however a sergeant from the narcotics squad, Melkowitz, shows up and questions them both, saying that Jack's friend Paul is also a heroin smuggler. Meanwhile, Michael has successfully stashed the heroin in Mary's diaper and has buried it with the other dirty diapers in the bathroom.
Peter appeals to his long-term girlfriend, Rebecca, to help him take care of Mary, but she refuses. One day Peter and Michael have Mrs. Hathaway babysit Mary and when they return home they find her tied up and unharmed and the place ransacked. Mary is found crying but also unharmed in a closet. In the middle of the night Jack (whose part in the film in Turkey was suddenly cut) returns and is nearly beaten up by both Peter and Michael, who think he may have broken in. They tell him about Mary and the drugs that his friend laid on them. Jack remembers that he'd acted in Stratford, England, in a production of The Taming of the Shrew about a year and a half previous, and had slept with Sylvia, however he wasn't aware that she'd become pregnant. Michael and Peter, angry at having to give up their free time and social lives solely because of Jack's irresponsibility, give Mary to Jack to care for, refusing to help him. Jack appeals to his own mother, who tells him that his life as a womanizer has finally caught up with him, and she refuses to help him as well.
Realizing they're caught between violent drug dealers and Melkowitz' narcotics squad staking them out, the guys figure out a way to escape the mess they're in: Jack will sneak out of the apartment dressed as a pregnant woman (with Mary hidden under his dress) and Peter and Michael will go to a designated place to hand off the drugs to the dealers. The police follow Michael and Peter but lose them when Jack picks them up in a taxi and heads off in the opposite direction. The meeting place is a construction zone for a skyscraper Peter had designed; Peter will hand off the drugs, Michael will film it on a video camera and Jack will trap the dealers in the construction elevator after they receive their package. Peter stalls a bit, getting the dealers to admit their culpability just as Michael, hiding in an aluminum vent cover, slips and falls. Michael and Peter make it to the elevator and begin to escape, however Jack throws the wrong switch, trapping them. The dealers descend in a second elevator but are trapped when Jack finds the right switch. Melkowitz and his squad arrive and arrest the dealers, taking the tape Michael had made for evidence.
Free and clear, the roommates take care of Mary for a few more months, working out a schedule beneficial to them all. One day Mary's mother, Sylvia, arrives; she has worked out a way to take Mary back to England and care for her while living with her parents. The guys are very upset at the new development, having grown attached to the baby. Sylvia leaves for the airport and the guys chase after her, arriving too late to stop the plane as it leaves. They return to their penthouse and find Sylvia & Mary at the door, who says she couldn't leave. The three men agree to let Sylvia move in with them.