According to the Washington Post, that assertion is at the core of the federal government's lawsuit challenging Arizona's new immigration law. So... the federal government has massively failed to deal with the problem of illegal immigration, but at least the failure is spread fairly equally among the states. Even though Arizona may only want to take responsibility for its its own problem, it can't do that without referring the cases to the federal government and straining and unbalancing federal resources. The courts are supposed to buy the paradox: Because the federal government can't do very much about a problem — or chooses not to do much — an individual state can't act either, no matter how bad things get within that state.
But let's think this through. I'm just trying to grasp what the argument is, so discuss this with me. Brainstorm. Argue. Consider this: The federal government has responsibility for immigration, and it has expressed, through written law and real-world efforts, an extremely lax policy toward illegal immigration. Given that federal policy and the supremacy of federal law, one could argue that it is not within the state's proper power to dictate a different policy and impose it on the federal government (by referring a lot of new cases of individuals violating federal law).
It will be interesting to hear how the lawyers for the federal government make their argument. Assuming it's not legally ridiculous, is it politically wise? To make it work legally, won't they have to own pathetically weak enforcement as a deliberate and important policy? Won't they have to be very clear that Arizona must shut up and accept the current situation? Who will get better political leverage out of this lawsuit — those who favor stronger enforcement of immigration law or those who favor leniency?