From my experience- being where you were for almost a year before I got my first job- I would say your classmates likely had some sort of in with the facility. Either they worked there before, they know someone who knows someone who knows someone and they rubbed elbows and kissed fannies. I would suggest, if you were friendly with anyone in your cohort, maybe calling them or shooting them an e-mail and asking them what their "secret" was. Maybe your resume isn't impressive enough, or eye catching enough. Maybe you need to do what I did (which is, now days either very smiled on or very frowned on, you have to take your chances) hand deliver a resume to the manager or whomever is in charge of the residencies. I hand delivered my resume to one nursing manager, and she interviewed me right then and there, it turned out she had someone else in mind, but she gave my resume to another manager on a different unit who called me a day or so later to have an interview. Of course several months back I did the same thing, and the manager was all but going to give me the job right there, but she had to contact the recruiter to get my information, and the recruiter went ballistic and called me and left me a really professional message (screamed it) and also ripped the nurse manager a new one too. So you have to approach with caution these things. It is likely that your resume is first passing through a recruiter, and often times their inbasket is overflowing with applications and resumes and some will just take the first 5 or 6 they encounter. Or the resume goes through a filtering system and sorts the resumes based on key-words. Look at the job descriptions or the requirements for the residency and make sure you insert those in your resume or cover letter.
These are all things I did, it took me a long, agonizing time. I researched on the web a LOT about dealing with recruiters and how to get one's resume noticed. Oh, one thing you could do, go back to your school, usually they have career services people who are happy to look at your resume and give you helpful hints on how to word things, format things and make it easier for computers/recruiters to process.
Hope this helps!