I'm not asking this question with a personal purpose (I don't have any fellowships and I wouldn't be greedy/unreasonable enough to abuse the system in this way), but I was still wondering out of plain curiosity:
Consider a graduate student who excels in the field and otherwise has many "favorable" attributes. In other words, if the student applies for any given fellowship, the chance that he/she will get it is very high. Given the cooperation of three professors being willing to write as many recommendation letters as the student wants, is there typically any restriction from the university or fellowship organization that would prevent the student from gaining an absurd amount of fellowships?