It has been now 2-1/2 years since I completed my Masters, but still not been able to get a PhD. I have got more than 50 rejections, in fact after 50 I have altogether stopped counting. It is really disheartening to know that you are not able to pursue something in which you've your complete heart. I am a non-EU candidate, and it already puts me in tight spot over funding issues. But for once I just wish someone would give a kind look to my application. I have good grades, passed M.Sc with merit and excellent references; but still what's it of use if you can't get what your heart really wants you to do. Quite frustrating must say.

you can write cover letter in 4 para. 1st para: who are you and why are you writing, 2nd para: why do you want to work with them i.e. check their background and check your academic interest and amalgamate them, 3rd para: why they should select you i.e. display your academic and non-academic abilities, 4th para: summarise all of the above, and thank them for their time and consideration.
Do not go beyond 1 page for cover letter. be brief and to the point, do not replicate your complete CV here, rather just choose couple of things from CV and make them sound interesting on cover letter.

I have been looking for a PhD (Human Genetics) from past 2 years with no success yet. I am a non-EU applicant and I am not self funded. I have graduated from my Masters (Medical Genetics) with merit and have excellent references. I have 90% in my GRE exams. Most of the times the replies to applications range from "..you're a high quality applicant, but the application process was highly competitive" to "..you're an excellent applicant but we received a lot of applications than we expected" and some not even bothering to reply at all...Which is really disheartening at times. I am currently looking for PhD in EU. All my applications, Motivation Letters and CVs are customized & specifically written for the given position. I have tried in Australia, EU, US & UK. So still wondering what exactly is the recipe to getting a position.
I guess sometimes it is just the timing and luck.