They certainly can. You have four years of NCAA eligibility, which you can extend if you are a graduate student. To compete at a Division I institution, student-athletes must achieve particular academic standards known as progress-toward-degree criteria. These include making certain grade-point averages and taking certain courses each year to remain eligible.
The only difference between being a student-athlete and an undergraduate student is that you're doing this while you're earning your degree. So the more involved you are in school activities such as sports, the more opportunity you have to earn points toward that degree.
Additionally, there are several sports (such as basketball and football) where you can participate with no impact on your graduation date. Even if you quit playing after one season, it wouldn't prevent you from graduating on time. The only thing that would happen is that you wouldn't be able to play again until after you completed your degree.
Finally, there are some institutions that have modified their rules to allow graduate students to compete in sports. Most schools limit how many hours per week or month you can spend participating in studies and sports. At these places, they may allow you to participate in certain number of games or practices without affecting your eligibility. Otherwise, you'd spend all your time studying or playing ball and not enough time doing both.
A Division I student-athlete who has remaining eligibility may compete in athletics while enrolled in graduate studies. A student-athlete must be within five calendar years of his or her initial enrollment in college to participate while in graduate school. In addition, students must maintain at least a 2.0 cumulative grade point average to remain eligible.
Students who complete their undergraduate work and intend to go on to earn a master's degree can do so by combining the two programs under the umbrella of "graduate sports." Many schools have moved away from this practice in an effort to provide equal opportunities regardless of class standing. Some universities specifically prohibit students from playing both intercollegiate sports and graduate programs. The use of academic all-stars is also becoming increasingly common as more players are able to balance athletic and academic responsibilities.
Many professional teams have scouting departments that monitor prospective graduate students as they attend school full time. These employees check players' grades during the year to determine their eligibility to play. If a student-athlete fails to meet the requirements after being granted a release from his or her previous institution, he or she cannot again compete for that team unless granted a waiver by the new school.
Graduate students are permitted to receive financial assistance through NCAA grants, scholarships, and loans.
Yes, but it must be earned in the classroom. While Division I sports scholarships are relatively straightforward to obtain—you must finish high school, complete 16 core courses, and have a GPA of 2.3—the requirements for student-athletes seeking non-athletic financial help are substantially more stringent. Non-revenue sports like football are allowed to offer aid, but only if it's intended to cover costs not covered by athletic fees. So, for example, if a player receives $10,000 per year in assistance, that money must come from another source (usually academics) because NCAA rules prohibit giving out competitive advantages through aid.
The value of the award must be reported to the IRS as income. It is also important to note that players can't receive any additional money or benefits beyond their grant-in-aid. For example, if a college football player receives $10,000 per year in aid, that player can't also receive $10,000 from someone else. The school must keep all aid over $2,500 separate from traditional funds.
In addition to the requirements for other students, athletes must maintain a 2.0 cumulative grade point average to continue receiving aid. If an athlete drops below 2.0, they will need to make up those lost credits to remain eligible. There is also some evidence that suggests playing time may be used as consideration for awards.