I suspect that the phone has been handled with, let's say, less than extreme care, being dropped a few inches, knocked around, etc. It sounds like a hardware problem to me. Possibly something as simple as a bad battery contact, but it could be more serious. (It could also be an intermittent power switch which, while being a really cheap part, requires a little disassembly to replace. Pressing the power switch for a few seconds [whether you do it or the switch does it because it's broken] will cause the phone to reboot, and if the switch isn't making contact when you press it, the phone won't power up.)
If the phone is just under a year old, see if they'll replace (or repair) it. Many stores will be a little lax on the "exactly one year" warranty, and will work with you even if it's a few days or a week over the one year point, but not much more than that - and some stores will look at the time you bought it and not honor the warranty if it's 5 minutes after it expired.