More than 57,000 X-rays taken at Tallaght Hospital in Dublin between 2005 and the end of 2009 were not reviewed by a consultant radiologist.
The hospital says that to its deep regret, two patients had a delayed diagnosis as a result.
One died last summer and another is undergoing treatment for cancer.
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Tallaght Hospital CEO designate Professor Kevin Conlon said he was 'stunned' to learn of the problem when he was appointed on 14 December last year and was only informed today of the case of the patient who died.
He said the X-rays would have been reviewed at the time by a non-radiologist, most likely another clinician but this was not acceptable.
In December, extra staff were appointed to clear the radiology backlog and to date, 34,752 of the 57,921 X-rays have now been reviewed and reports finished.
The hospital hopes to have the 23,169 remaining X-rays reviewed by May. The X-rays were for various tests including cancer, heart checks and orthopaedics.
The hospital says the HSE was informed of the problem in December.
Prof Conlon said what had happened was totally unacceptable and arose from systemic and process failures.