Even though a battery is new, it should still be charged when put into service. In most cases, the vehicle will do this this normally, but those driven short distances may never fully charge. Completely discharging a new battery (any battery for that matter), is not good for it unless it is a deep cycle battery (usually not found in autos). This cycling causes "sulfation", which then causes the battery to not take a charge. However in your description, I don't think this is what was happening as these processes take a fair amount of time, and repeated depletion. I wish I had a good explanation for you problem, but don't. I would recommend getting the charging system evaluated. On several of my vehicles, I run this cheap voltage indicator I got at Radio Shack that plugs into the cigarette lighter. I has 3 leds (Red, Yellow and Green). The car should show a green LED after starting and later sometimes a green and yellow combination. Straight yellow is less then 12.5 volts, and yellow with red is less than 11.5 volts. These colors tell you the charging system is not working correctly. Mine are over 15 years old, so I'm not sure they are sold any more but you might check. Good Luck, please let us know the out come.

Jerry