In the 1850 census James and Elizabeth Barnett are listed in Limestone County, Texas. They appear at the bottom of page 360. At the top of page 361 there are seven Longbottom family members listed ranging in age from 25 to 9. They appear to be in the same household as James and Elizabeth Barnett.
On the top of page 364, the children of James and Elizabeth Barnett are listed. They appear to be part of the Harden household which appears at the bottom of page 363.
At first glance it appears that the Barnett children were not living with their parents at this time. But another question arises: who are the Longbottoms and why are they living in the James Barnett household? A search for Longbottoms in the 1850 census in Limestone County, Texas found a Longbottom couple (ages 55 and 54) at the bottom of page 358 – could they be the parents of the Longbottom children?
I think that somehow the census pages got mixed up before they were numbered and that the Barnett children were in reality living with their parents, not the Harden family.