Here's my checklist:

Is the green light on? If not, check with the FA to see whether the outlet is working

Does the green light turn yellow when you plug in the device? if yes, then it's drawing more power than the outlet provides, so the safety kicks in. As a work-around:
a) If possible, remove the battery from the laptop. This will ensure your laptop is powered. Save often, as an accidental kick to the cord will shut down the laptop.
b) Plug in the laptop in sleep mode until the battery is fully charged, then turn on the laptop and it'll remain charged.