If your charger is capable of a little less than 1200 mA (I've never seen an N3 draw more than that from any 5 volt source), and you're using more than the charger is capable of putting out, simple math tells you that you're not charging the battery, you're just discharging it slower. (Charge = amount put in - amount being used.) If you're using a cable other than the original, even if the charger is able to put out enough current, if the wires in the cable are really thin, the voltage drop in the wires keeps the phone from getting a good charge.
Other than that, the only problems would be a bad battery or a relatively high resistance short somewhere in the phone.
Use the original charger and the original cable and if it still happens ask Sprint for a warranty exchange. (Try it with not too much running. Turning on all the radios - wifi, data, Bluetooth, GPS - will add a bit of drain, and running half a dozen apps at the same time, especially with the screen on and bright, could cause a bit of discharge even with the OEM charger and cable.)