It is possible.
EDIT - Okay, here's just one example, in Illinois, for all of you who have no idea what you are talking about:
In an unpublished opinion, the Illinois Supreme Court has held that the termination of parental rights does not extinguish a child support obligation unless the child is being adopted. The court relied on the language of the Illinois statute, which refers to termination of parental obligations as to “a child sought to be adopted.” The court stated: "We conclude that, after the entry of an order terminating parental rights, where the child is not adopted, [the Act] applies as its plain language indicates, only where the child is 'sought to be adopted'…. To hold that this language is of no effect, as respondent urges, would render the language superfluous or meaningless." The court rejected the appeals court’s interpretation of the “sought to be adopted” language as including any child 'available for adoption', noting that the legislature could have specifically used this language.
The law on this varies from state to state. What state the asker's cousin is in, we don't know.
In re: C.S. Stover