What typically limits how many fellowships a PhD student can receive?
Generally the organizations offering the fellowships have some rules regarding accepting other sources of funding concurrently.

For example, the Ford Foundation fellowship:


  A Ford Foundation Fellow may not accept remuneration from another major fellowship or similar external awards while on Ford Foundation Fellowship tenure; however, limited supplementation from institutional funds, educational benefits from the Veterans Administration, or educational incentive funds may be received concurrently with Ford Foundation support. Fellows receiving support from multiple sources are urged to discuss and negotiate the terms of the funding with all funding agencies and their university.


Universities also typically have policies regarding concurrent fellowships. For example, at the University of a Illinois:


  Fellows are prohibited from holding two major awards concurrently (fellowship, traineeship, grant, tuition payment award or comparable support from any government agency, state, federal or foreign, or from any foundation, corporation or similar organization). Any award offer should be reported immediately to the Graduate College Fellowship Office, where the determination will be made whether the two awards may be held concurrently.