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That type of problem could be indicative of a bad battery, a bad cable, a broken cable connector on the phone or moisture damage. The first thing I'd do is try different cables, including the USB cable and the A/C charger. Try plugging them in without the battery in the phone and see if you can turn it on. Then replace the battery and see if it is any different.
Is there any chance the phone got wet? That could cause a lot of problems.