Does Cingular have an internal number to dial from the phone for supprt?
T-mobile has 611. It doesn't matter if the card is active or not. 611 will always reach the t-mobile customer care line.
If cingular has a similar type number for their phones, you might try it. If you still get the chime, there could be bigger problems.
As far as making a few calls before it went funny, I really don;y know. Is this a transfered card from another phone? New from the store? Have you checked to make sure the SIM card hasn't wiggled itself loose?
Those are my best guesses. Good luck.