The Standish sanitarium, located in the resort town of Sparkling Springs Lake, has fallen on hard times. Young Judy Standish (Maureen O'Sullivan) inherited the family business, along with its debts and mortgages, and is determined to keep going despite the fact that local businessman J.D. Morgan (Douglass Dumbrille) is pressuring her to sell out. Morgan needs the sanitarium property in order to expand his racetrack and resort hotel empire, and is conniving to have the sanitarium fail, aided by Judy's oily business manager, Whitmore (Leonard Ceeley), who is secretly in cahoots with Morgan.Judy's last hope is to request financial assistance from her most well-heeled client, the wealthy widow (and hypochondriac) Mrs. Emily Upjohn (Margaret Dumont). To Judy's dismay, Mrs. Upjohn storms into the lobby and announces that she is leaving. The incompetent medical staff has had the nerve to tell her there is nothing wrong with her. Mrs. Upjohn yearns for the ministrations of Dr. Hugo Z. Hackenbush, since "I never knew a thing was wrong with me before I met him." Overhearing this exchange, Judy's loyal man-of-all-work Tony (Chico Marx) announces that Dr. Hackenbush will soon be arriving. He then rushes off to telegraph Hackenbush offering him a job. As Tony observes, "If she want a hack-in-de-puss, she gonna get a hack-in-de-puss."Judy's fiancé Gil Stewart (Allan Jones) is hoping to make it as a singer, but in the meantime has come up with a quicker plan to earn enough to save the sanitarium and marry Judy. He has sunk all his savings into purchasing a race horse called Hi Hat, to be ridden by an eccentric local mute jockey named Stuffy (Harpo Marx). A couple of realities intrude on Gil's plan. Hi Hat's practice times are mediocre, and Gil has no money to keep up with the horse's expenses, causing the local sheriff (Robert Middlemass) to come after the horse to repossess it for the unpaid feed bill. Fortunately, some sleight-of-hand by Tony and Stuffy keeps the sheriff at bay for the time being. Gil is also dismayed that he has a sure thing tip for a race that afternoon, but no money to bet. Tony reassures him that he will get some money by finding a sucker.In Florida, Dr. Hugo Z. Hackenbush (Groucho Marx) is treating one of his clients, a horse, when the telegram arrives with the job offer. Aware that they can put a veterinarian in jail for treating people, he observes that he can also be put in jail for not paying his rent and has nothing to lose. Soon he arrives at Standish Sanitarium, where Judy appoints him chief of staff at the behest of Mrs. Upjohn, who has agreed to pay Hackenbush's salary and has hinted that she may help the sanitarium financially. The rest of the medical staff, and the scheming business manager Whitmore, are suspicious of Hackenbush's credentials, but he manages to double talk his way around them, and then, at the sound of the bugle call, heads for the racetrack. On his way to place a bet on Sunup, Hackenbush encounters Tony in the guise of an ice cream vendor. Tony offers a hot tip, which turns out to be in code, which a variety of code books and reference works are needed to interpret. After spending a wad of cash on useless books, Hackenbush winds up with a losing horse while Tony bets the profits on the winner - Sunup!Whitmore attempts to verify Hackenbush's credentials with the Florida Medical Board, but Hackenbush manages to divert the calls, to Whitmore's frustration. In order to keep an eye on Whitmore, Tony brings Stuffy into the sanitarium for treatment. During Hackenbush's examination of Stuffy, Tony discovers that Hackenbush is really a horse doctor. Hackenbush wants to flee, but Tony insists that he stay to try to help Judy. At the water carnival, where Gil is the featured vocalist, Hackenbush escorts Mrs. Upjohn but is distracted by a voluptuous blonde named Flo (Esther Muir), with whom he performs an energetic rumba and arranges a late-night rendezvous. Tony and Stuffy invade the bandstand and treat the audience to a comic musical performance. Afterwards, Stuffy overhears that Flo has been hired by Whitmore to lure Hackenbush into a compromising situation. He manages to convey this information to Tony, and the two rush to the rescue. Flo arrives at Hackenbush's suite at the sanitarium for a late night supper. Hackenbush tries to romance Flo, but Tony and Stuffy keep interrupting, first as house detectives, then as decorators hanging wallpaper. At the crucial moment, Whitmore leads the indignant Mrs. Upjohn to Hackenbush's suite, but Flo is safely hidden inside the sofa under a pile of wallpaper, and Mrs. Upjohn redirects her anger to Whitmore.Stuffy hides the horse in a room in the sanitarium. In a further attempt to discredit Hackenbush, Whitmore has brought in Dr. Steinberg (Sig Ruman), a pointy-bearded Austrian physician, to examine Mrs. Upjohn. Steinberg declares that he has never met a healthier woman in his life, to which Hackenbush retorts, 'You don't look as though you ever met a healthy woman." Pressured into demonstrating how he conducts an examination, Hackenbush, aided by Tony and Stuffy, manages to postpone doing so by repeated hand washing and other diversionary tactics. The examination eventually erupts into mayhem when Stuffy sets off the fire sprinkler, and the trio escapes riding Hi Hat.Hackenbush, Tony, Stuffy, and Gil hide out with Hi Hat in a remote barn next to an African-American community. Judy arrives with blankets and Gil reassures her about the future. The locals are enjoying their day of rest with song and dance. The kids are fascinated by Stuffy, whom they identify as the angel Gabriel; Stuffy plays a tin whistle and leads them in a dance. A girl (Ivie Anderson) sings "All God's Children Got Rhythm." Then the sheriff arrives along with Morgan to confiscate Hi Hat. Upon hearing Morgan's voice, Hi Hat bolts and clears a tall hedge with ease. Judy and Gil realize that the horse is a jumper, not a sprinter, which means he's more suited to steeplechase racing, and that Hi Hat's animosity toward Morgan can be used to advantage.The day of the big steeplechase has arrived, but all will be lost if Hi Hat cannot make the post time. Staking out the track, the sheriff once again wrests control of Hi Hat, but then Judy gets the horse back by staging a road accident.Meanwhile, Hackenbush, Tony and Stuffy come up with various and hilarious stratagems for delaying the start of the race. To Morgan's dismay, Stuffy appears at the last moment on Hi Hat, ready to race. After seeing Morgan's photo, Hi Hat easily makes the first jump, but then Stuffy loses the picture. Hackenbush comes up with an alternate plan - he finds a microphone hooked up to the public address system and gets Morgan to talk into it, thus energizing Hi Hat. The jockey of Morgan's horse has been instructed to play dirty, and does so. Eventually Hi Hat and Morgan's horse wind up falling into a big mud puddle. They recover and race to the finish, with Morgan's jockey edging out Stuffy. Judy and Gil's disappointment is palpable, but when Morgan arrives to claim the prize, the winning horse reacts negatively to him and Stuffy realizes that it is really Hi Hat - the jockeys had switched horses in the mud. The race is won, the sanitarium is saved, Judy and Gil can marry, and most amazing of all, Hackenbush proposes marriage to Mrs. Upjohn!