Parents said the poor state of St John's Primary in Portobello contributed to the accidents, and two families have now launched compensation claims against the city council.
The council, meanwhile, has ordered "finger guards" to be placed in all doors in the school following the latest accident.
The father of five-year-old Elliot Telfer hit out at the "appalling" state of St John's following his son's accident last week.
Mark Telfer said Elliot had a "lucky escape" after the youngster's fingers were trapped in a fire door leading to the toilet block. Mr Telfer, 38, visited the scene of his son's accident the same night and was "astounded" at the condition of the school, which is one of five awaiting refurbishment or replacement under the city's "wave three" programme.
Mr Telfer, who runs a storage business, said: "Part of the problem with the door is that it needs to be replaced.
"I was absolutely appalled with the state of disrepair at the school, especially in the toilets where Elliot trapped his fingers.
"He was very lucky because he could have lost his fingers.
"The doctors at the Sick Kids said he was very fortunate.
"Boys are always jumping around, you can't wrap them in cotton wool, but I was flabbergasted when I found out what had happened to him. I can't fault the school staff and headteacher, but they're there to teach kids, not watch them 24/7 in case fire doors that don't have adequate stoppers close on their fingers."
The other two incidents happened within the last year.
In the most serious case it is understood that a young girl's fingertip had to be re-attached after it was trapped in a door. The two families involved are seeking compensation though Mr Telfer is not and said he just wanted to ensure it did not happen again.
David Manson, chair of the school's parent council, said he asked for assurances on the safety of the doors in the school after the previous incidents, which he described as "horrendous".
He said: "Some of the doors are an accident waiting to happen, they are great big heavy doors and some of them are designed to swing shut.
"The state of the school is such that things that should have been done decades ago are still waiting to be done."
A council spokesperson said: "All the doors which have fire door closers in schools are inspected annually to ensure they are closing properly and meeting health and safety regulations.
"Additional finger guards are now being fitted as an extra precaution to ensure that this does not happen again."