Their early work, as yet unpublished, involved looking at the GP records of 3.5 million people in the UK.
They say people who were obese but who had no initial signs of heart disease, diabetes or high cholesterol were not protected from ill health in later life, contradicting previous research.
A summary of their study was discussed at the European Congress on Obesity.
The term "fat but fit" refers to the alluring theory that if people are obese but all their other metabolic factors such as blood pressure and blood sugar are within recommended limits then the extra weight will not be harmful.
In this study, researchers at the University of Birmingham analysed data of millions of British patients between 1995 and 2015 to see if this claim held true.
They tracked people who were obese at the start of the study (defined as people with a body mass index of 30 or more) who had no evidence of heart disease, high blood pressure, high cholesterol or diabetes at this point.
They found these people who were obese but "metabolically healthy" were at higher risk of developing heart disease, strokes and heart failure than people of normal weight.
Dr Mike Knapton, from the British Heart Foundation, said: "It's not often that research on this scale and magnitude is able to clarify an age-old myth.
"These findings should be taken extremely seriously and I'd urge healthcare professionals to take heed."
He added: "Previously we used to think that being overweight led to an increase in heart attacks and stroke because it raised your blood pressure or cholesterol.
"What was new from this study for me is that it showed that people who were overweight or obese were at increased risk of heart disease even though they may have been healthy in every other respect.
"Just being overweight puts you at increased risk of heart attack and stroke."
But the study has not appeared in a scientific journal and, as such, it will not have gone through a number of checks by other academics to judge whether it is scientifically sound.
It is difficult for example to know how well other influential factors - such as diet, lifestyle or smoking - were taken into account.
This makes it hard for scientists to see how clear-cut the conclusions are or gauge how big any increased risks of ill-health might be.
According to the British Heart Foundation, the normal heart health advice applies - not smoking, eating a balanced diet, exercising regularly and limiting alcohol intake - can all help keep people healthy.
Dr Knapton added: "This is not about laying the blame at individuals though.
"This is a wake-up call for planners, local councillors, food manufacturers and the government to make sure we can make healthy choices more easily."
Dr Rishi Caleyachetty from the University of Birmingham, added: "The priority of health professionals should be to promote and facilitate weight loss among obese persons, regardless of the presence or absence of metabolic abnormalities."
He added: "At the population level, so-called metabolically healthy obesity is not a harmless condition."
Other studies suggest it might just be possible to be fat and have the right genes, for example, to remain fit.
For example research published in 2012 appears to suggests it is possible for people to buck the trend and be fat and healthy if they have no metabolic diseases.
Published in the European Heart Journal, researchers suggest people who are obese yet physically healthy are at no greater risk of heart disease or cancer than people of normal weight.
But other experts point out that the way scientists measure fatness and fitness makes this a tricky area to study and could make some of the more tantalising results invalid.
Tom Sanders, emeritus professor of nutrition and dietetics, King's College London, says a major weakness of the Birmingham study is that it uses definite cut-offs to decide when someone has high blood pressure or high cholesterol for example.
Instead, he argues that it is too simple and not accurate to use such definite values to decide whether someone is healthy.
And other studies have suggested that it is not always the amount of fat that matters but where the excess fat is carried on the body that can affect fitness and health.
For example, weight around the middle may be more damaging than weight distributed evenly around the body.
Overall, experts say it is important to not just focus on what you see in the mirror or on the scales - exercise and healthy eating can help boost wellness, no matter how much a person weighs.
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