I keep getting alerts stating startup disk is full when it's not I keep getting alerts saying that "your start-up disk is almost full".  

I have an early 2015 13 inch MacBook Pro running macOS 10.11.6 with a 128MB SSD that has 63GB free according to About this Mac > Storage.  I have created two other standard user accounts for my wife and son.  I typically see this issue after coming home from work after they have been using it.  If the alert appears, About this Mac > Storage will indeed indicate that all storage has been used, the bulk being of type: Other.  When this happens, any open apps are sluggish, frequently showing the spinning wheel cursor and preventing saving.

Restarting the system solves the issue.  If I had to guess, it seems like some program is hogging all disk space for swap space.  The only app with which I am unfamiliar and didn't install myself is Discord which my wife installed for my son.

Any thoughts on what could be causing this phantom disk usage?
Do they ever get the error? Find out who's account is actually the one causing it. Then log in as them and have them use the computer until it happens again. Once the drive is full, double click on the "Macintosh HD", switch it to "List" view, and go to View > Show View Options > Calculate all sizes. That will let you track down the mystery 63 GB and see exactly what is using it.