Laptop suddenly started randomly turning off and has great trouble starting. Once started, easily shuts down again

This is a year-old Dell Latitude 5500. At around 95% battery, it suddenly shut down. Pressing the power button causes the power lights to flash, sometimes for a fraction of a second, sometimes for a few seconds. Sometimes, the keyboard lights turn on, and it boots to Desktop. I would say that it boots around 1 in 20 attempts. The first time this happened, it almost immediately shut down again. The second time, it was able to stay on, until I closed the screen, and it should down again. On one occasion, it showed a time of day not set error , with several menu options. Another time, it said tha

Question by Al2110 on Super User, licensed CC BY-SA. We show a short excerpt and link to the original thread as credit.