Actually, my step Father attempted to disown me. No love lost there. He's a very bad person.
Ironically, a biological parent can legally disown their children - but an adoptive parent is bound by law to maintain the legal condition.
Until a person is 18 years old in the U.S., there are a whole slew of legal issues that can invoke the burden of responsibility upon the parent.
A friend of mine was disowned by his parents because he very actively sought to make their home lives difficult. His parents were tired of receiving phone calls from the local authorities and were burned out on spending days in courthouses and counselling sessions. Once they legally disowned him, and he was on his own (actually in a boys home) they were free of the problematic issues they had because they were no longer legally responsible and now they could actually call the police on him. Years later - they reconciled and he recognized that had his parents not acted and put him out on his own, he'd likely still be a non productive member of society and be hiding under the ignorant protection of his parents. Today he has a good relationship with his parents and siblings, is married and has kids, he's a part owner of a auto repair shop. Not glamorous by any stretch of the imagination, but he's a very responsible Dad who is very in love with his kids. Not to say that all hardship stories have good endings... Unfortunately, more the contrary is true.
Personally, I cannot imagine disowning my kids. Even if I dislike the paths they choose in life and the interests they give themselves to I cannot imagine turning my back on them. I have seen and I do understand situations where children make life difficult for the parents though. I said that I cannot imagine turning my back on them... not that a situation might arise in the future where that thinking could be challenged.