A number of patients were given up to double the recommended dose of Pfizer vaccine at the Queen Margaret University drive-in vaccination centre on Saturday, after a dilution error.
The issue affected fewer than five patients, but NHS Lothian was unable to confirm the specific figure due to confidentiality rules.
Above-recommended doses can cause serious side effects, and those patients affected were notified by the health board and told to seek medical help in the case of any problems.
One man given a higher dose said the incident “shouldn’t have happened” but stressed that while his family had been concerned, he was not badly affected.
The 68-year-old, who was receiving his second dose of vaccine, experienced just an ache in his arm for a couple of days, which he also felt after his first vaccination.
“I got the jag, which was okay, and then they phoned me two hours later and said there was a possibility that the doses hadn’t been diluted properly,” he said.
"She said it wouldn’t do me any harm, but that you can get side effects, and it was just to watch out for it really.”
The man said his family had been concerned.
"These things shouldn’t happen, you should get a proper dosage,” he said.
But he added that it “would have been worse” to have been given too little of the vaccine by mistake instead of too much.
“I’m 68 years old and I’m just happy to get a vaccination, and that’s the two of them done now,” he said.
Professor Alex McMahon, NHS Lothian nursing executive director, said: “NHS Lothian is aware of a very small number of cases where the Covid vaccination dose administered has been higher than the optimum recommended amount.
"This has not caused any serious concerns or health issues, however it can make the recipient more likely to experience short term side effects such as a sore arm or cold like symptoms within 48 hours.”
The first known overdoses of Pfizer vaccine, given in Germany before Christmas, led to four care home workers being hospitalised after up to five times the recommended dose.