I think that the most productive thing to do in this situation is to concentrate on yourself. There's no way of knowing why this other person is succeeding and it's nothing that you have any control over. Comparing yourselves is not productive- I would focus on what you can do to get hired, and worry less about her.
In terms of "getting noticed", sometimes it's a matter of timing. Some places won't look at resumes if they are not hiring, so sending them a resume at the wrong time of year may mean that it doesn't get looked at.
Do you know someone at the theatre that can put in a good word for you? I used to send my resume to the same theatre over and over with no response. Then, I changed one of my references to an SM who had worked at the theatre before and the next resume I sent in, I got an interview. It may be a matter of finding a personal connection to that theatre.
It may also be that they are looking for someone with a particular set of skills- reading music, experience with classical theatre/musicals/new plays/whatever. It's impossible for me to say what that particular company is looking for- each company is different and values different things in their employees. Some theatres prefer to hire people who have interned there and work their way up. See if you can find out what they look for when hiring- look at an old job posting, ask someone who's worked there before, etc.
Keep building experience. Is the "top theatre" that you are hoping to work at out of your experience level? Maybe there is an in-between step that you're skipping.
Sometimes it just takes time- but you can always be actively working to make yourself more marketable to the places you want to work at.