The school had contended that the state's charitable immunity act protected it from liability in sexual abuse lawsuits brought by former students.
In 2001, Kenney filed a sexual abuse lawsuit against a former priest at his high school.
They became clear when a judge ordered it to release files in abuse lawsuits in January 2002.
In January 2002, the diocese settled 11 sexual abuse lawsuits brought by 16 plaintiffs for about $14 million.
His name figures in many of the church documents released in abuse lawsuits in Boston.
Depositions of the cardinal scheduled for this week in an abuse lawsuit have been postponed.
Over 500 sexual abuse lawsuits are still pending against the archdiocese, the country's largest.
Potential settlements in sexual abuse lawsuits also loom over the archdiocese.
One-time costs, including settlements resulting from sexual abuse lawsuits, accounted for $7.7 million of the estimated shortfall.
Also at issue is how the church will deal with the hundreds of sexual abuse lawsuits filed against it.