"New" Galaxy S4 won't turn on

I'll be getting tech support for this, but I have to wait on that till Monday and I figured I may be able to get some help from you guys.

I recently replaced my aging and now buggy S2 while switching to a new phone carrier for a cheaper plan. My phone of choice was a Galaxy S4 (factory refurbished) and it just came in yesterday. I turned it on (70% power from the factory charge they ship it with for the first time setup), let it do it's first time setup, realized that the charger connection was a bit spotty and it didn't seem to really be charging, so I turned it off and let it sit on the charger over night. I went to bed with the red charge light on and the green battery icon showing on the screen (the one that shows when the phone is off but charging). I woke up and the phone is completely unresponsive. No charge screen icon. Won't turn on while plugged in or just using the battery. I've tried soft resetting it, no joy. I don't know phones nearly as well as I do desktops/laptops, so I don't know if there is any more troubleshooting I can do. Come Monday, I'll be calling about a replacement if need be, but maybe myself and other users may get some help from this thread. Thanks for any help.

Did you try physically removing the battery then putting it back in?

Yes, that's what I meant by soft reset.

So... I tried to turn my phone on again just now and it did. This is probably the 100th attempt, but now it's on for no apparent reason.