Flexible Pell Grant for 21st Century Students Act This bill amends title IV (Student Assistance) of the Higher Education Act of 1965 to allow an eligible student to receive more than one Federal Pell Grant award in a single award year. An eligible student is a student who: (1) has already received a Pell Grant during an award year and is enrolled in an eligible program for one or more additional payment periods (e.g., a summer term) during the same award year, (2) continues to meet Pell Grant program eligibility requirements, and (3) is accelerating progress toward a degree or certificate. An eligible student's total Pell Grant amount must not exceed 150% of the annual maximum Pell Grant for that award year. An institution must notify a student that any period of additional Pell Grant receipt counts toward that student's lifetime Pell Grant eligibility period (currently 12 full-time semesters). If an additional Pell Grant spans two award years, then an institution must determine the award year to which such Pell Grant applies. The Department of Education must provide to each Pell Grant recipient an annual status report that includes certain information related to such student's remaining lifetime Pell Grant eligibility period.