The poverty advocates’ stratagem of legerdemainically misleading the public about Rhode Island’s welfare system by focusing on TANF numbers has been repeated for years. It caught my attention back in 2004, and assuming that they still apply, some of my points from then may serve to supplement Andrew’s posts (here and here).
TANF dollars aren’t the only cash handouts available in Rhode Island:

SSI, or Supplemental Security Income, is a federal program that provides monthly cash payments to people in need. SSI is for people who are 65 or older, as well as for blind or disabled people of any age, including children. …
The state of Rhode Island adds money to the federal payment. The single payment you get in the beginning of each month includes both the federal SSI payment and your supplement from Rhode Island.

Rhode Island adjusts TANF payments based on other income less than other states do (from my old post):

The folks not included in this analysis are those who have some form of other income. To understand why this matters, consider what looks to be the comparable program in Massachusetts. … That single mother of two will, indeed, receive $633 per month in Massachusetts if she doesn’t live in subsidized housing. However, her income (after certain deductions) is directly subtracted. So, suppose she gets $200 from some other source. In Massachusetts, her monthly cash gift would be $433, with $633 remaining her monthly income.
In Rhode Island, on the other hand, her base benefit would be $554, and the first $170 of additional income isn’t counted. Moreover, the cash benefit is only reduced $1 for every additional $2 of income. For the woman making $200, that would result in a $15 reduction. So, this same woman who was capped at $633 in Massachusetts would take home in Rhode Island: 554 + 200 – 15 = $739. And in fact, the percentage of those who benefit from FIP who are working rose from 13.7% in 1997 to 21.3% in 2003, which has been at least part of the reason for [the decrease in Rhode Island expenditures for cash assistance.]

But the real bloat is in child care and healthcare, which helps to explain the difficulty in tracing the degree to which RI’s welfare system draws the poor to our state:

In fact, all Rhode Island households earning no more than 225% of the federal poverty level are eligible for child care subsidies, with copays ranging from $0 to $48 per child per week. For a family of four … that means annual income of $42,413. According to the U.S. census, the median household income in Rhode Island for 2000 was $42,305. Rhode Island apparently considers half of its families to be “low income.”
From a taxpayer point of view, it’s also interesting to note that, to encourage child care providers to accept poor children, they get fully paid healthcare. And healthcare opens a whole ‘nother stack of taxpayer bills. Every family receiving cash payments from the government is eligible for it. Every family with income up to 185% of the federal poverty level ($34,873 for a family of four) is eligible. And every child under 19 and pregnant woman with household income of 250% FPL is eligible. If the household makes less than 150% FPL, the insurance is free, otherwise there are relatively tiny monthly payments of $61, $77, $92.

ADDENDUM:
One mild adjustment to Andrew’s reference to the five-year limit for TANF: That’s the limit that the federal government puts on the program, but states may opt to shorten it.