RAND researchers studied more than 450,000 recipients of New Jersey's Tuition Aid Grant (TAG) -- the nation's most generous state-funded grant program per state resident college student -- to explore whether getting larger amounts of grant aid led to higher graduation rates for students at varying income levels and attending two-year, four-year, public, and private institutions. The data set covered school years 2012-2013 through 2019-2020 at 52 colleges and universities. Students received from $1,000 to $13,000 per year in TAG awards, plus $600 to $6,000 per year in federal Pell Grant aid. Grants help students cover tuition and living expenses and do not need to be paid back. On average, TAG aid increased the rate of on-time bachelor's degree completion at public universities, the sector where TAG has the most recipients and covers the largest percentage of tuition costs. An additional $1,000 in aid over four years led to a 2.6 percentage point increase in the graduation rate from a sample average graduation rate of 35 percent, a significant increase and higher than the average effect found in prior studies of other aid programs. In community colleges and at private colleges and universities, the effect of TAG for its lowest-income recipients on four-year graduation outcomes was positive and marginally statistically significant.