Thank you for providing the detailed description of the problem. You have mentioned that the Num Lock light flashes for 7 seconds and then the computer powers off. This does not indicate any common failure, however from your description it seems the system has power but is not able to power on completely because one of the components may be failing. If the keyboard status lights are lit in any of the following sequence they might indicate a system problem in Inspiron 6400 Laptop:
1. On-Flashing-Flashing = No Ram
2. On-Flashing-On = Ram Fault
3. Flashing-Flashing-Flashing = Motherboard
4. Flashing-On-On = CPU
5. Flashing-Flashing-On = Video
6. Flashing-On-Flashing = LCD
7. On-On-Flashing = Mini PCI Card
8. Flashing-Flashing-Off = Keyboard
9. Flashing-Off-Flashing = USB
10. Off-Flashing-Flashing = Modem
I have mentioned the keyboard light status in the sequence starting from left (Num Lock, Caps Lock and Scroll Lock) and the failing component is listed against them. Do reply if you observe any of these indications.
Also you have mentioned that you read from other forums and found that one of the Memories was failing. You can try reseating the memory modules in the machine to try a fix for the problem; also you can try one memory module at a time and check if the problem still exists. You can follow the link below to see how to reseat the memory modules:
Do reply with results.