A US couple who gave their children Nazi-inspired names should not regain custody of them, a state appeals court in New Jersey has ruled, citing the parents' own disabilities and the risk of serious injury to their children.

The state removed Heath and Deborah Campbell's three small children from their home in January 2009.

A month earlier, the family drew attention when a supermarket refused to decorate a birthday cake for their son, Adolf Hitler Campbell.

He and siblings JoyceLynn Aryan Nation Campbell and Honszlynn Hinler Jeannie Campbell have been in foster care.

A family court had earlier determined that there was insufficient evidence that the parents had abused or neglected the children. That decision was stayed until the appeals court could review it. On Thursday, the three-judge appeals panel determined there was enough evidence and that the children should not be returned.

The panel sent the case back to family court for further monitoring. A gag order remains in place and the parties refused to discuss the decision.

Heath Campbell said last year that he believed the children were taken because officials felt they were in "imminent danger".

He accused the state of removing the children because of their names and said government officials were relying on unproven accusations made by a neighbour and by an ex-wife who charged him with abusing her years ago.

The children's names and the birthday cake were not mentioned in the ruling, but the court found that there were numerous other reasons that proved the need for continued protection services for the children.

According to court records, both parents are unemployed and both suffer from unspecified physical and psychological disabilities.