The GOP chant of the 1884 election: "Ma, Ma, where's my Pa? Gone to the White House, ha-ha-ha." A real toe-tapper.

What was really hidden was Cleveland's secret operation for cancer of the jaw. After being diagnosed, he gathered a team of doctors plus a dentist and announced he was going fishing on Long Island Sound on a yacht borrowed from a millionaire friend. He was operated on successfully and fitted with a rubber prosthesis. Although hints of his bout with cancer were published during his presidency, he got away with it. The full details weren't revealed untin after his death.

The American Medical Association did a review of secret illnesses of presidents, and how they were treated. Cleveland's was the earliest case, and they concluded that he was the only president who had competent medical care. Other cases were shocking:

Wilson's physician, Cary Grayson, a Navy doctor, had only one year of medical school. He and Wilson's wife conspired to keep his debilitating stroke secret--in effect, she ran the presidency for more than a year.

Harding's physician was a naturopath of doubtful credentials. He diagnosed Harding's extreme cardiac illness as "indigestion," leading to his death.

FDR's physicians deliberately kept his atherosclerosis secret from everyone. His blood pressure regularly was 300/200. They treated him with digitalis. In his last months, he was unable to concentrate for more than an hour a day, and often zonked out in the middle of sentences.

Eisenhower's physician, Howard Snyder, a ENT, deliberately expunged a chapter of Ike's medical record while he was a general that reported a moderate heart attack. Grateful Ike made him the White House physician when he was elected. Ike had another, major, heart attack in Denver in 1955. Snyder showed up two hours after being warned by Mamie and found Ike in extremis. He waited 12 hours before contacting a local hospital, then allowed Ike to get dressed and walk under his own power to the car that drove him to the hospital.

As for JFK, who may have been our chronically sickest president, here is a relevant selection from Sally Bedell Smith's "Grace and Power":

JFK’s life necessarily revolved around the vast responsibilities of the office, but less obviously his schedule included unusual accommodations to his myriad ailments. His principal physicians, Janet Travell and Admiral George Burkley, worked in a room on the ground floor, and he had an array of specialists in regular contact by telephone that included an allergist, endocrinologist (for his Addison’s disease), gastroenterologist (for colitis), urologist (for urinary tract infections resulting from venereal disease), and orthopedist (for his degenerative spine).
JFK’s regimen of strong medications requires strict oversight and constant calibration, especially the corticosteroids for his Addison’s. To treat his chronic back pain, Travell gave him daily injections of the local anesthetic procaine. Kennedy also routinely took Cytomel (for thyroid deficiency); Lomotil, Metamucil, paregoric, Phenobarbital, and Trasentine (to control the diarrhea from his colitis); testosterone (to increase energy and boost weight following bouts of colitis); penicillin (for urinary tract flare-ups); Fluroinef (to increase his ability to absorb salt, which Addison’s depleted); Tuinal (for insomnia, a side effect of the cortisone); antihistamines (for an array of allergies); vitamin C; and calcium supplements (to substitute for milk products, which exacerbated his colitis). As a precaution against triggering an Addisonian crisis, JFK’s doctors boosted his cortisone when he faced stressful situations such as speeches and press conferences.
Dr. Max Jacobson, a Manhattan physician to the rich and famous, was called “Dr. Feelgood” for his mysterious injections laced with amphetamines. Several days before his first debate with Nixon, Kennedy had secretly visited Jacobson’s office for an injection to relieve symptoms of fatigue and muscle weakness. Afterwards, he told Jacobson he felt “cool calm and very alert.” On Tuesday, May 23 [196l], Kennedy sneaked Jacobson into the White House for four days of amphetamine shots “to relieve his local discomfort” and “provide him with additional strength to cope with stress,” recalled Jacobson. Concerned about the impact his [back] injury could have on his stamina in the Paris and Vienna meetings [meeting DeGaulle and Khrushchev], Kennedy asked Jacobson to join him for the trip.
The Kennedys had barely a half hour to settle into the American ambassador’s residence (in Vienna) before JFK’s first meeting with Khrushchev. Almost immediately, Max Jacobson was directed to Kennedy’s room. “The meeting may last for a long time,” JFK told him. “See to it that my back won’t give me any trouble when I have to get up or move around.” Jacobson administered his injection, and a tanned and youthful Kennedy was next seen bounding out of the front door “like a bronco-buster sprung from his chute,” wrote Time.