Health concerns that have dogged Democratic presidential nominee Hillary Clinton for months burst to the forefront Sunday when she abruptly left a 9/11 commemoration ceremony due to illness, and her campaign later reluctantly acknowledged that she had been diagnosed with pneumonia days earlier.

Mrs. Clinton is weathering a deluge of speculation about her health, including claims that she suffers from debilitating conditions ranging from seizures and dementia to Parkinson’s disease that would impair her service in the Oval Office.

She has laughed off questions about her health, dismissing them as “conspiracy theories.” Her personal physician has vouched for her overall good health, and independent medical experts have disputed some of the speculation about her presenting symptoms of a progressive, debilitating illness.

The incident at the 9/11 ceremony in New York, however, whether a fainting spell or a case of being “overheated and dehydrated” as her physician stated, only elevated concerns that Mrs. Clinton is hiding something about her health.

The suspicion also underscores the widespread distrust of Mrs. Clinton and the belief that she is deceptive.

Mrs. Clinton and Republican nominee Donald Trump were on hand Sunday for ceremonies to memorialize those who died 15 years ago at the World Trade Center in lower Manhattan.

Mrs. Clinton — a former first lady, U.S. senator from New York and secretary of state — was whisked away early after becoming ill, according to her campaign.

A witness to the incident, which occurred away from the view of the public and the traveling press, told Fox News that Mrs. Clinton stumbled off a curb, her “knees buckled,” and her shoe fell off as she was hustled into a van.

The Clinton campaign downplayed the health scare.

“During the ceremony, she felt overheated so departed to go to her daughter’s apartment, and is feeling much better,” Clinton spokesman Nick Merrill said in a statement.

Mrs. Clinton, who turns 69 next month, emerged from the New York apartment building about two hours later. She was smiling and waving without any assistance before she climbed into a black sport utility vehicle.

The fresh scrutiny prompted the campaign to later release a statement by Mrs. Clinton’s physician, Dr. Lisa R. Bardack, explaining that the candidate had been diagnosed with pneumonia Friday during a follow-up visit for coughing spells related to allergies.

“She was put on antibiotics, and advised to rest and modify her schedule,” Dr. Bardack wrote. “While at this morning’s event, she became overheated and dehydrated. I have just examined her and she is now re-hydrated and recovering nicely.”

In the pre-antibiotic era, pneumonia was among the leading causes of death, and it can still often be fatal when it strikes children, the elderly or people in otherwise-poor health.

Mr. Trump, who turned 70 in June, has fueled interest in his rival’s health by repeatedly saying that she lacks “mental and physical stamina.”

Dan Pfeiffer, a former communications director with the 2008 Barack Obama presidential campaign, downplayed the diagnosis, taking to Twitter to say “every candidate I have ever worked for has gotten sick on the trail and worked through it because you can’t take days off in a close race.”

A senior Clinton aide told The Wall Street Journal on Sunday that Mrs. Clinton was reconsidering a scheduled airplane trip to California on Monday to appear on “The Ellen DeGeneres Show.”

Speculation that Mrs. Clinton’s health is worse than she is admitting dates back to her fall in December 2012, when she sustained a concussion and was hospitalized. Doctors discovered a blood clot in her brain, which was treated and resolved, according to Dr. Bardack.

The concussion resulted in double vision that required Mrs. Clinton to wear special eyeglasses. She wore them during testimony in January 2013 before a Senate hearing on the terrorist attack on the U.S. diplomatic outpost in Benghazi, Libya.

Mrs. Clinton’s behavior and demeanor on the campaign trail added to the speculation:

• She appeared to stumble and had to be helped up stairs by Secret Service agents.

• She avoided interaction with reporters, including going more than nine months without a formal press conference.

• She suffered a prolonged coughing fit during a Labor Day speech in Cleveland, and later that day couldn’t stop coughing when greeting reporters on her airplane, forcing her to retreat to the private cabin at the front of the plane.

Some of Mrs. Clinton’s supporters and several liberal-leaning journalists angrily dismissed the week of whispers about Mrs. Clinton’s health as sexism.

TV and radio personality Dr. Drew Pinsky said earlier this month that he was “gravely concerned” about Mrs. Clinton’s health and the treatment she is receiving, saying the care described by her physician was “1950-level care.”

Dr. Pinsky, a board-certified internist more commonly known as Dr. Drew, said he was particularly concerned about her double vision from the confusion.

“That is brain damage, and it’s affecting her balance,” he said. “Now, clearly, it hasn’t affected her cognition, but tell us a little more about that. That’s profound.”

Shortly after he made the remarks, CNN announced it was canceling his show “Dr. Drew on Call.”

Dr. Ted Noel, an anesthesiologist from Orlando, Florida, reinforced rumors that Mrs. Clinton has Parkinson’s disease in a video posted on the political commentary website VidZette. He outlined many of Mrs. Clinton’s purported health issues and video of her displaying odd mannerisms to make the case that she is concealing an advanced stage of the neurological disorder.

The doctor’s evidence included Mrs. Clinton’s history of falling dating back to 2005, coughing fits, odd hand gestures while speaking, repetitive head-nodding and her bug-eyed expression during the balloon drop at the Democratic National Convention in July.

“Early in 2016, Hillary’s problems started to be really obvious. Videos of odd physical movements and brain freezes became common,” Dr. Noel says in the video. “This is consistent with the progression of Parkinson’s disease into its later and more troublesome stages.”

Stanley Fisher, a neurologist and one of the nation’s leading experts on movement disorders, including Parkinson’s disease, disputed the diagnosis offered by Dr. Noel.

“I’m not a Hillary fan at all. But this video was truly a hack job by a half-informed physician, and I am ashamed that this doctor would even try to do this,” said Dr. Fisher, the medical co-director of the Saint Luke’s Marion Bloch Neuroscience Institute in St. Louis.

“It is my opinion, by observing her in multiple videos, that there is no evidence to suspect that she has even early Parkinson’s, let alone advanced,” he said, noting the obvious manifestations of the disease as seen in pubic figures such as actor Michael J. Fox, former U.S. Attorney General Janet Reno and the late boxing champ Muhammad Ali.

He said that it was not unusual for a 68-year-old woman to occasionally stumble or get lightheaded, especially if she keeps a rigorous schedule, as does Mrs. Clinton. But he said that not enough information was provided to fully assess her health.

“It is possible that she is having some kind of issue, but again, nothing to suspect some kind of progressive or degenerative disease,” said Dr. Fisher. “But it doesn’t mean she is healthy. That I want to make clear.”

• Tom Howell Jr. and Dave Boyer contributed to this report.

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