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George Bush unfurled the ‘Mission Accomplished’ banner on the deck of one of our aircraft carriers a few years ago, and that didn’t turn out quite to be true,” said Mr. Bonner, whose union represents more than 12,000 non-supervisory agents.

He said it will take at least 25,000 agents to properly secure the southern and northern borders, and Mr. Bush could have gotten closer to that goal if he had made it a priority earlier in his administration.

Mr. Bush came into office hoping to pass a landmark immigration bill, but was caught between competing forces in his own party. While some Republicans wanted to pass a bill in order to secure a flow of immigrant workers for the future, most Republican lawmakers said his plan to grant citizenship rights to illegal immigrants amounted to amnesty.

Democrats, meanwhile, were split, with most backing Mr. Bush while some said his legalization plan was still too harsh on illegal immigrants and skewed too heavily toward business interests rather than family unification when it came to future immigration.

Mr. Chertoff urged Congress to stick to basics of the agreement that senators reached last year.

“I think what would be a mistake is if one side tried to dramatically skew it one way or the other,” he said.

He also warned Mr. Obama against splitting up the Homeland Security Department, which was created in 2003. In particular, he rejected calls to split off the Federal Emergency Management Agency, arguing that the recent terrorist attacks in India show the importance of coordinating emergency response and law enforcement.

“The reporting a couple of days ago about Mumbai was the fire department, the emergency managers and the police came on the scene and, according to the news reports, they were not well coordinated. That, in a nutshell, is the argument for keeping all this stuff consolidated. When you look at a crisis or an emergency, you cannot stovepipe your emergency response and your police response.”