http://www.nyc.gov/html/nycha/html/assistance/income.shtmlThe problem with most public housing in NYC has more to do with the terribly designed neighborhoods around the developments than the income limits. They built giant blocks of apartments with no reasonable storefront or business space around it, creating relatively isolated high-density apartments without allowing physical spaces for neighborhoods to develop. You need high density and mixed use, apartments for the sake of having housing alone is a recipe for a city's stagnation.
Attracting a lot of poor people to a city is a good thing - it means that there are incentives attracting people to the city, which can be a decent indicator of the economic health of a city. But the blocks of public housing (particularly between 112 and 115, but really throughout East Harlem, and the Lower East Side) is a textbook example of failed 'tower in a park' housing concepts that were destined to be an economic sinkhole from the start.
They were premised on outdated 19th century principles that wanted to limit tenement style health problems without recognizing that lack of green space wasn't what was causing the health issues in high density environments, it was bad infrastructure and limited access to clean food/water.