Lily Denby, six, was taken to Wrexham Maelor Hospital after she hurt herself playing at home with twin brother, Harri.
Lily's Dad Darren, 46, says they only X-rayed Lily's wrist, so didn't see the break nearer to her elbow.
Doctors said she'd suffered only soft tissue damage to her left arm, and sent her home with advice to take Calpol for pain relief.
She returned two months later, after injuring the same arm when she fell at a play area. On that second visit, medical staff at the Maelor's emergency department not only found she'd broken her arm - they also uncovered the first fracture, and put her arm in plaster.
Darren - a father-of-three from Hope, Flintshire, said: "We went to A&E in June and we were in there for quite a few hours and she was sent home with a diagnosis of soft tissue damage. It seemed pretty busy in there and we waited around for an hour and a half before she had an x-ray. They then spent five minutes with us later on and said it was soft tissue damage.
"Not long after I let her go swimming but she had to come out as she was in discomfort – but I naturally trusted the doctors. A lot of the time over the next few months she was in pain when she was putting her top on. We went the second time after she fell off the log and the nurse said 'that looks broken to me'.
"They x-rayed her arm and x-rayed it in exactly the same place and said 'we're not happy with this' so carried out a further x-ray. She was in casualty all night and the doctors said it was broken and it was also broken the first time – so I got them to write a letter to confirm it.
"It was only by pure luck that the first break healed OK. She happened to have chicken pox for a week and didn't move much."
After the original diagnosis, in June 2013, Lily's dad let her carry on swimming and playing football for Llay United FC.
Although she occasionally complained of pain in her arm, the fact she had broken it was not discovered until the second accident at a play area in Gresford in August.
Darren complained to Betsi Cadwaladr University Health Board (BCUHB) via the Betsi Cadwaladr Community Health Council (CHC), but has accused the health board of making excuses.
In its response the health board apologised for failing to identify the initial fracture but said it believed it didn't contribute to her second arm injury.
Mr Denby wrote a complaint to BCUHB via the CHC on October 1, but he did not receive a response until last Monday (13 Jan) – more than three months later.
"They took months to reply," said Darren. "They haven't explained to me why the second time was different. They were brilliant the second time. They mention about not being eligible for compensation but this is not about trying to get compensation.
"I want a proper acknowledgement and I want reassurance that this won't happen again."
In her response to Darren, Angela Hopkins, BCUHB executive director of nursing, said: "Having investigated this concern, the health board is satisfied that there is no qualifying liability within the definition provided by the regulations as Lily was treated appropriately for her presenting condition at that time.
"However we would like to apologise that Lily did suffer a fracture in her arm which was not identified as she did not show any signs of this at her initial presentation to the emergency department. Miss Rae (consultant in emergency medicine) is confident that Lily not receiving treatment for this fracture did not contribute to her subsequent injury."
In a statement yesterday, a BCUHB spokesman said: "Due to patient confidentiality, we are unable to comment on individual cases.
"However, we can confirm that a concern has been received via the Community Health Council, which has been investigated and a response sent. We are sorry to learn that Mr Denby is unhappy with this response and we would be more than happy to meet with him to discuss his concerns further."