Before 9/11, the trial run for the attack was the World Trade Center bombing. The key figures in that attack, from Ramzi Yousef, the nephew of 9/11 planner Khalid Sheikh Mohammed, to the Blind Sheikh, whose ambitious plans involved bombing New York bridges and tunnels, asked for political asylum.
Ramzi Yousef showed up with a fake passport and claimed to be an Iraqi refugee seeking political asylum. He was released on his own recognizance and told to show up for a hearing. He never did.
His co-conspirator, Ahmed Ajaj, went on filing for political asylum even while under arrest for his role in the World Trade Center bombing. Their pals, Mahmud Abouhalima, who routinely told passengers that "In Islam, the people are the weapons", and Mohammed Salameh, were illegal aliens who had applied for amnesty as agricultural workers. But they didn't go on to pick lettuce or blueberries.