Some background on Faisal Shahzad - the suspect in the recent New York City bomb attempt - is here, including some information on his immigration status. In roughly chronological order (quoted portions as in the article):

* End of 1998 or very beginning of 1999: "granted a student visa, which requires a criminal background check".

* Jan. 16, 1999: arrives in the U.S.

* 2000: gets a bachelor’s degree in computer science and engineering from the University of Bridgeport.

* Unknown time between 2000 and 2004: marries Huma Mian; she currently lives in Pakistan.

* "In January 2002 Mr. Shahzad obtained an H1B visa, a coveted status meant for highly skilled workers and good for three years, with a possible extension. Records show that Elizabeth Arden, the cosmetics giant, applied for a visa around that time for a job similar to the one he had there in 2001, arranged through a temporary employment agency called Accountants Inc., according to a timecard found in his trash. Officials at the cosmetics company refused to comment."

* 2004: petitions to become a legal permanent resident (i.e., "green card")

* Spring/Summer 2004: "George LaMonica, a 35-year-old computer consultant, said he bought his two-bedroom condominium in Norwalk, Conn., from Mr. Shahzad for $261,000 in May 2004. A few weeks after he moved in, Mr. LaMonica said, investigators from the national Joint Terrorism Task Force interviewed him, asking for details of the transaction and for information about Mr. Shahzad. It struck Mr. LaMonica as unusual, but he said detectives told him they were simply “checking everything out.”"

* 2005: gets an MBA from the same college.

* October 2008: applies for citizenship.

* April 17, 2009: attends his citizenship ceremony.

Obviously, not all immigrants from Pakistan or other countries that have links to terrorism are going to turn out like he did, however it would be a very good idea to do even more thorough background checks on those who might have other agendas. It's probably not too likely that he was suddenly radicalized.

Note also that at the current rate that they do background checks, it would take 5 to 10 years to do FBI background checks on 10 million people. Any amnesty plan will thus have lightweight checks; for instance, the Reid-Schumer-Menendez plan would do a lightweight check, gave them permission to stay here, and then send them on their merry way. Only years later would another check be done. Obviously, that wouldn't be enough to stop someone like Shahzad.

UPDATE: From this:

Sources tell CBS News that would-be Times Square bomber Faisal Shahzad appeared on a Department of Homeland Security travel lookout list - Traveler Enforcement Compliance System (TECS) - between 1999 and 2008 because he brought approximately $80,000 cash or cash instruments into the United States.