The New Jersey State archives has a copy of Catharine Noe's death certificate. She died in 1887 in New Jersey. Parents are listed on her death certificate as Rice Carrington (b. Litchfield, CT) and Abigail Fox (b. Litchfield, CT). Catharine is listed as b. Litchfield, CT and having lived in New Jersey for 50 years, dying at age 75.
Catharine married James in 1845 - I believe that's after James Atwood Noe and Thomas Augustus Noe were born, which points to James having a first wife and Catharine being the second, if indeed James Atwood and Thomas Augustus are his children. Thomas Augustus died in CA in 1914 and his death certificate notes maiden name of mother and birth place of mother as "unknown". James Atwood died in Oklahoma in 1922 and his death certificate notes maiden name of mother and birth place of mother as "don't know."
There's also an article in the 31 July 1841 Connecticut Courant seeking information on Catharine Carrington, parents Rice and Abigail Carrington. The article notes that for five years prior to leaving Litchfield, Catharine lived with the family of William Demming and that she'd been last heard from 2 years ago.