You only need one nomination to get into an academy. However, some nomination sources can increase your chances. I am speaking in terms of USMA, which is what I know. I don't have enough knowledge of the other academy processes to comment on theirs. What I mean is this... each nomination source (each senator, congress person, etc) will have one cadet (at a minimum) each year attend West Point. Then, West Point allots for a certain number of cadets from ROTC nominations, military nominations, etc. West Point historically does not receive enough qualified applicants in those areas, so some of those slots are then filled by someone who received a congressional nomination. This means that historically receiving a military nomination COULD increase the chances of gaining acceptance versus a congressional nomination. I stress could because every year is different and each candidate still has to meet certain minimum qualifications.
I hope this makes sense. Feel free to PM me if you have a question.