No, if they are not a dependent, you would not usually claim them. They might qualify you for Head of Household.
For the purposes of the Head of Household filing status, a qualifying person is a child, parent, or relative who meets certain conditions, listed below.
The conditions are stricter than those for claiming a dependent; for example, you might be able to claim a roommate as your dependent, but never as a qualifying person for Head of Household status.
A qualifying child would be:
A child that is too old to qualify as a child might be able to qualify as a relative for Head of Household.