Among those in their first and second year of work this fell as low as 5 per cent.
Dr Lailah Peel, chair of BMA Scotland’s Junior Doctors Committee, said the "shocking” survey results were worse than she had expected.
“When you see that about 5 per cent of our first year junior doctors are getting their breaks on time, that’s pretty appalling,” she said.
The survey of 330 union members over the festive period also showed just under 70 per cent of junior doctors not receiving their rotas on time, six weeks in advance.
It comes as 121 medical and dental staff absences were reported in Scotland last week – the second-highest figure since June 2020.
Dr Peel said the problems had been exacerbated by the pandemic, but existed before.
Junior doctors are often working an entire shift without a break, taking just a few minutes off, or having to work through their breaks, she said.
This has led to concerns for the wellbeing of patients and doctors.
“There are times where I've been on shift over the last few months where I've felt like patient safety is suffering, or patients definitely aren't getting the treatment we would like them to be getting,” said Dr Peel, who works in accident and emergency.
“I think we're all feeling like it's a matter of time before something will happen.”
She added: “We can't keep going like this. You can’t keep working at your absolute full capacity, working at 120 per cent every single shift, every day, for this kind of period of time.
“Something is going to fall apart, and whether that’s the individuals, whether that's the system, or whether that's going to be the individual patients that suffer, it’s worrying either way.”
Dr Peel stressed there were safety mechanisms to protect patients from mistakes made by tired doctors.
“As doctors we do everything possible to prevent that from being the case,” she said. “But we're human, unfortunately, we will all make mistakes at different points.”
The union is most concerned about the newest doctors, who started work during the pandemic and report getting fewer breaks than senior colleagues.
"These junior doctors are the consultants and the GPs of the future, and if you work them this hard they just aren't going to make it to the end of their training,” said Dr Peel.
She has also noted an increase in aggressive behaviour and verbal abuse from patients, many of whom are frustrated by long waits at A&E and difficulty accessing care elsewhere, such as from GPs who are also overloaded.
A&E performance times in Scotland hit a new low this week, with almost a third of patients waiting more than four hours and over 2,000 waiting more than eight hours in the week to January 9.
First Minister Nicola Sturgeon admitted the NHS is under “significant pressure”.
She told MSPs on Tuesday: “As is reflected in today’s A&E waiting time figures, the past couple of weeks have probably been the most difficult the NHS has ever faced.”