The parents of Robert Benton, who will attend the funeral of their child today, said he might be alive if he had received speedier treatment once he developed breathing difficulties.
The parents, Julie Benton and Tim Dawes, of Rowley Regis, West Midlands, say they are considering legal action against West Midlands Regional Health Authority. The child was first taken to Sandwell hospital, West Midlands, near the family home, at about 12.45pm last Wednesday after a cough had developed. The family left after 15 minutes because it seemed so busy.
They then took him to a second hospital, the Good Hope hospital, 10 miles away in Sutton Coldfield.
Once there Robert was X-rayed, given a blood test and put on oxygen. Staff said the boy had a collapsed lung and needed specialist help at Birmingham Children's hospital, nine miles away.
But when the child arrived there by ambulance there was no bed for him and he was then taken another five miles to the city's Heartlands hospital, where he died on the operating table.
A West Midlands Regional Health spokesman said all four hospitals involved were concerned about the tragedy it had ordered an inquiry.
David Clifton, the boy's uncle, said: "What Julie should have been doing now is planning her son's birthday party, not his funeral.
"She put her faith in four hospitals and they let her down."
Miss Benton, 23, said: "He was just not given priority yet he had one collapsed lung and fluid in the other. No one can convince me that baby should have died. He was healthy."
The family say they were told Robert died from tracheobroncho malacia - a condition in which the windpipe does not grow in tandem with the rest of the body, causing breathing difficulties.
Miss Benton, a bank worker, and Mr Dawes, 32, were told their son had died at about 9pm - some eight hours after he was first presented at a hospital.
They have since been told that Robert was born with a weakness in the connections to his lungs which had not been detected but the family say the condition had never been mentioned to them before despite previous breathing complaints.
Mr Clifton, speaking for the family, said: "We are concerned that if this condition had been discovered earlier in the day they could have done something. We are not satisfied with the way it was handled.
"Nothing will bring the child back, but the parents have a right to know the best was done for him and, if not, to ensure it will be done in the future."
John Spellar, Labour MP for Warley West, said the health authority inquiry should report quickly and make its findings public.