HARTFORD — A couple whose two young sons were taken away by the state for several months in 2000 over a doctor’s disputed abuse opinion lost a bid on Monday to reinstate a lawsuit against the state child welfare agency.

The state Appellate Court rejected an appeal by former Newtown residents Michele and Craig Sienkiewicz, whose baby son had multiple bone fractures. The couple sued the state Department of Children and Families and agency officials for an undisclosed amount of money, citing emotion pain, humiliation, legal fees in fighting the agency and other reasons.

The couple was challenging a lower court dismissal of its lawsuit in 2014. A Superior Court judge’s ruling included a finding that the state was shielded from the lawsuit by government immunity. The Appellate Court, the state’s second-highest court, did not decide the merits of the lower court ruling but said the couple’s lawyer failed to adequately address the legal issues in the appeal.

The child welfare agency took temporary custody of the boys in February 2000 after one of them was found to have 20 rib fractures and two knee fractures shortly after birth and the couple’s pediatrician reported the injuries to the agency as a case of suspected abuse, court documents said.

Experts for the couple would later testify in court that the injuries occurred at the time of birth and were the result of either medical personnel pushing on Michele Sienkiewicz’s stomach or from pulling the child out of the uterus during the cesarean section.

Child welfare agency officials sent the baby to a foster home and the couple’s older boy, who was then 5 years old and showed no signs of abuse, to live with an aunt, according to court documents. It also applied to terminate the couple’s parental rights to both children, citing the baby’s injuries and the couple’s inability at the time to explain them.

Weighing the conflicting medical testimony, Superior Court Judge Dennis Eveleigh, now a state Supreme Court justice, ruled in favor of the couple in May 2000 and ordered the children to be returned to their parents immediately. The child welfare agency appealed the ruling to the Appellate Court but dropped the appeal later that year.

The Sienkiewiczes first sued the child welfare agency in 2003, and the court battles have continued off and on since.

It’s unclear if the Sienkiewiczes plan to appeal Monday’s ruling to the state Supreme Court. They and their attorney did not return phone messages seeking comment on Monday.

Superior Court Judge Dale Radcliffe, in a 2008 ruling in the case, called what happened to the Sienkiewiczes a “nightmare.”

“That they discovered the financial and emotional resources to regain custody of both children … seems insufficient to erase the emotional trauma to which they were subjected,” the judge wrote.