This sounds like the common problem of broken pins in the cartridge slot.
Look down into the slot and see if any of the small gold pins are missing. They should all be in a perfect row, with no gaps.
This was very common on the original DS model, and early DS lites.
Late model DSi's have a dust flap that prevents you seeing into the slot to look for damaged pins.
If a single pin has broken off then i am afraid the unit is not worth trying to repair. The pin itself in unrepairable, the slot type extremely hard to source a direct replacement, the entire board from Nintendo is more than a new unit, it isn't covered by the warranty as it is classed as abuse/neglect, and no-where will even try and fix it without charging you an assessment fee for being able to do nohing to it.
Most likely bad news to you i'm afraid...
Drew
DrewKrew, March 2011