Does not really increase your chances per say, but having more than one nomination simply pools you with a different slate of candidates to compete with. So in a sense, I don't believe one nomination has anything to do with the others, but rather they are all separate and it really doesn't matter how many you have, but rather how competitive you are on each of the slates.
For instance, your senatorial noms may be really competitive and you may end up ranking 2nd, or 3rd on those slates and therefore be placed in the national pool. You do, however have one last shot with your congressman who may happen to have a less competitive pool of candidates, allowing you to win the slate and therefore earn the opportunity of an offer of appointment.
So in conclusion, the more nominations you have, the more chances you have to be presented before the admissions board and compared to other slates of candidates. But, as you probably understand, nothing is ever set in stone, but by acquiring three nominations you have given yourself the best chance possible to earn an appointment. Best of luck to you.
USMC8
USNA Class of 2017 Appointee