I have a new Samsung Galaxy Centura (StraightTalk / Tracfone, purchased at Walmart Dec2013).
I discovered that it doesn't want to power on when it is plugged into my laptop's USB port for charging (not sure if it makes a difference, but battery charge was below 10% prior to charging). It only shows the battery charging display.
I used to have an old iPhone prior to this Samsung, and with the iPhone I used to have to wait for approx 5 mins when charging a nearly depleted battery before the phone was usable again. (BTW, this was very annoying if the phone died in the middle of an important call due to a low battery.) At first I assumed that the Samsung was exhibiting a similar behavior.
With the Galaxy I have discovered that the power on function (when charging) seems to work "when it wants to". (...my success rate has been approx 1 out of 10.)
MY SOLUTION (seems to work consistently for me thus far) is to simply unplug the phone's charging cord, and then press the power on button.
With this method the phone powers on 1st time every time.
Once the phone is working I then plug it back in to resume charging.
I'm hoping that this solution persists. (& I hope this simple solution suggestion helps someone else.) :-)