This issue seems to come up a lot with Acer laptops, so I wanted to share what I found:
I opened up the bottom of my Aspire E15. I found this little button which seems to be an interlock that shuts off the laptop when the bottom access cover is removed. You can see the button in the middle of this picture. It's the small switch inside the square hole.

The inside of the bottom access cover has a square post that looks like it's supposed to keep the button pressed. Here's what the square post looks like.

This Youtube video isn't mine, but it shows how to open and replace the back access cover, and you can see the little button. The Youtube video is called "Acer Aspire E15 E5 575 521F RAM/Memory upgrade (E5-574-53QS)"
I suppose I could add tape or something so the post presses more on the button, but I didn't add tape to the post. I simply opened and reclosed the back access cover which seems to have fixed the problem. If the problem returns, I will add some tape so that the square post pushes a little more on the button, but it's been 10 days now, and so far my laptop is working fine.