You might have a failed battery. When the battery is really flat it will shut the phone down and not allow you to turn it on - until you get the big fat battery gauge on the screen. When/if you get a new one there is a Battery Monitor Widget that will keep stats for you, look for "BMW" on the market.

(PS. I have an Optus branded one. Do Telstra fill it full of battery sucking, persistently annoying crapware like Optus does? Optus have reined it in a little. At least you can uninstall)

 

That sounds like a hardware failure tbh, if a known-good battery also doesn't work. It's still well within the maunfacturer's warranty period so should be repaired/replaced without charge.

Good luck.