That sounds like a problem with power delivery on your tablet/laptops motherboard.
Ruling out the adapter:
If the Adapter Light comes on this typically means that the adapter is properly converting AC (from your wall) voltages to the desired DC voltage. If you have a multi meter you can always check that you are getting the proper voltages. These can be seen on the adapter itself. Or if you have a spare adapter check to see if the same symptoms happen. If they do the adapters are working properly.
What might be occurring on your laptop:
If your laptop is off and has no power, and you try to connect the adapter. The light on the adapter will go out when you connect it/ try turning the laptop on. The laptop has a failed component that is making it draw to much voltage, which in turn will trigger the tablet/laptops auto off feature to prevent further damage to the components. So this will prevent your battery from charging and allowing your device to turn on.
What you can do:
1. If you have a spare laptop adapter try using that to check if your laptop turns on.
2. (Best Solution) If your product is still under warranty you should contact ACER.
3. Otherwise you are left with replacing the motherboard/finding a repair shop.
4. I don't think batteries are removable on tablet/laptops so I cannot tell you steps for this.