2011 MacBook Pro won't turn on
It is likely the issue explained on this page on apple.com. If so, it is a problem with your GPU.

Symptom wise the "trial and error" in trying to start up your computer corresponds to my own experience, which I have also seen online. I personally also didn't see any actual graphics glitches before experiencing my first trouble startup up my computer. Only when my computer was already in a pretty bad state, did I see them.

I am quite sure I have this issue on my other computer, because using a third party program gfxCardStatus I was able to avoid crashes when I switched to "Integrated Only", which is the internal GPU. That avoids using the GPU that has the problem.
Note that you may have to act quickly to get reimbursed for a repair. The final date is Februari 27 for this issue.

More info

http://mbp2011.org/