This SONY VAIO (VGN-NR38M) was brought to me with the problem of "Not Powering up" and sure enough it looked dead. However the power LED would light and when the power button was pressed, the Battery and HDD LEDs would light briefly. You could hear the HDD begin to spin up but then stop. This indicated initially there was an internal power problem. 

However, I've come across similar problems with laptops that turned out to be a faulty RAM Stick so decided to check these before opening up the whole computer. Luckily with these older laptops, access to the RAM modules and sometimes the HDD is via a small removeable cover underneath. I opened the panel and looked at the RAM modules, there was nothing physically obvious with them, not surprising really. I removed the power, removed one stick of RAM, reapplied the power and tried to start the laptop, again it failed. I repeated the process after refitting the stick and removing the other and was rewarded with a power up boot screen, thus indicating the second RAM module had failed, probably going short and taking down a power rail.

The two modules were 512MB and I checked that I had a suitable replacement. I also had a couple of 1GB modules and asked the customer if she would like the RAM upgraded to 2GB, which she was happy to do. She also advised that the laptop had had a few unexpected crashes and boot failures recently, which was a precursor to the total failure. Obviously the module was on the point of failure.

Replacing the modules and rebooting I was rewarded with a full boot and operation. The customer was very pleased to get her laptop back working and with an upgrade!