Q: "Disk almost full" message

try Restart

then Disk Utility>First Aid

Exactly the same here as well, though on a Mac Mini Late 2012.  Have never seen it prior to upgrading to macOS Sierra.  More than 300Gb free on a 500Gb HDD so there is no issue with space.  Disk Utilities reports no errors (though I haven't tried running it from Restart Recovery mode).

I've just put it down to yet another spurious glitch

Type the command "df -h" (minus the "s) in a Terminal window and paste the output here.

I Have the same problem. It is not a memory problem from the file end As I too have significant space left. I was on the beta of Sierra. I may do a complete wipe and install clean OS.

Same here. Almost 500GB of free space, but since updating to macos Sierra this keeps popping up for me.

Same here on a Mac Mini with over 900GB free space on a 1TB fusion drive. It's only started happening since I upgraded to Sierra.