Most likely the problem is the charger. They are a piece of crap! Make sure to take the charger with you so they can check it.
The cables are notoriously fragile. The advice to take the cable AND charger in to B&N is good. They may change it outright if you have both pieces.
One clue can be the LED colors on the cable. It should be off when disconnected from the NOOK, yellow(ish) while charging, and green when fully charge. On one defective cable I had, it would glow red when disconnected, and a half-hearted green while charging. They changed it at the store with minimal fuss.
You might take your receipt along to emphasize you just got the unit.
Thanks for the information. I went to B&N yesterday and they ended up replacing the entire unit. I am really impressed with the service I received.