What if there is more than one doctor?
From the games instructions found here:


  When completing College, you do not draw an Experience Card; instead
  you score an automatic salary increase of $20,000, and select a degree
  as you leave. If you are the first player to pass through College,
  select any degree from the specialized degrees offered, and check it
  in the "Education" column of your score sheet. However, each degree
  may be awarded only once. Thus, if you are the second player to
  graduate from College, and the first graduate selected the Science
  Degree, your choice is limited to the remaining three, and so on for
  the first four players. If you are the fifth or sixth player to pass
  through College, you may not score any of the specialized degrees, but
  do score the regular College Degree and the normal salary increase of
  $20,000. Any College Degree, specialized or regular, may be used for
  entering Big Business. A player selecting the degree in medicine
  becomes the Doctor. He or she receives payments for players in the
  Hospital; the Doctor does not have to pay upon landing in the Hospital
  and may move out of this space on his or her next turn. (Note: If the
  Doctor is instructed to go to the Hospital he or she must go
  immediately even though they do not follow the instructions printed on
  the square when they get there.) No player may go to College more than
  once.


I'm not an expert at the game by any means, but I would infer from the sentences I've highlighted above that, in fact, there can only be one Doctor in any one game of Careers.