I happen to know quite a bit about immigrants in the Washington area. I was an interpreter for an organization providing immigrants with pro-bono legal advice. The stories I came across were heartbreaking.
Far from gaming the system, these immigrants were so afraid of authority that they did not dared apply for any benefits they were entitled to. We had to guide them through the resources available in the area, starting with free English night classes.
For the most part they were submissive, beaten down people, used to being taken advantage of by authorities and employers alike, and totally ignorant of their rights. For instance, I met a Dominican man, a skilled carpenter heading a six man crew rebuilding the Pentagon after 9/11. He designed and supervised the building of concrete frames used for columns. Yet, he was paid a third of what an American apprentice carpenter in his crew got. He learned that from the apprentice himself, who urged him to get legal help.
I also worked as a ride-along interpreter with the local Police. We there was an area they called The Black Hole of Calcutta, a street so dangerous that police cruisers had to patrol it in sets of two. It was full of drug dealers and gangs - all Americans. Gradually, Hispanic immigrants started renting houses on that street, and whole families moved in. The area changed so much that in a couple of years we could drive down the street in a single cruiser, with windows rolled down and people waving at us.
It may be different in other cities - I just know what I saw for myself.