Can a college student play football and basketball?

The simple answer to this question is: Possibly

The detailed answer is: There is a lot involved in playing sports in college. Not all colleges offer football and not all colleges offer basketball. If a college does offer these sports, usually the general student population cannot just suddenly decide they want to play and sign up. Coaches go to high schools across the country and recruit the best players available. This is even the case with Junior Colleges- not just the big Universities.

It is possible for a student who wasn't recruited to try out for a team, these students are called "walk-ons". If the student is talented enough, he can beat out the recruited players and earn a spot on the team.

It is also possible for an athletic student to play multiple sports in college. It isn't the easiest thing to do- the student will have plenty of competition- but if he is the best player available for both teams and has coaches who are willing to work with him, it is definitely a possibility. Basketball and football have the added problem of overlapping season. However, while it is unlikely there have been students known to make the teams and excel in both.