The only help I have found for this is to delete building blocks and let Word regenerate new ones which will wipe out all of the custom built tables. I also found this
Fix it for me

Migrate your custom templates using the easy fix tool.
The easy fix tool fixes all Office programs at the same time. You don’t need to specify the template location in each Office program.
However, the easy fix tool doesn’t help you to set a new location. It only migrates to the Custom Office Templates folder under My Documents. To set a new location, see the "I'll fix it myself" section below.



Does this work in this situation? We are trying to get people upgraded but there are some who will be on 2010 right up until just before April deadline for the drop support by MS.

Your custom quick tables are stored in templates. The default storage template is Building Blocks.dotx in Word 2010. You could have stored them in any template.
Where can Building Blocks be stored?

When you say they will not open in Word 2016, what, exactly, happens when you try?

This error message comes up, Word cannot open this document template. And then gives a path (C:\Users\...\16\~$ilding Blocks.dotx)

But if I click on it a couple of times, the customized tables show up. They were created with Word 2010.

Close Word
Find this locking file, which Word should not be able to open, and delete it.
It will be in the same folder as the Building Blocks.dotx file for Word 2016.

Just wanted to verify it would be called Built-In Building Blocks and the file size is 3620KB as opposed to the Building Blocks file which is 39KB. Will this delete the custom tables that were created?

NO!

Building Blocks.dotx is what contains your custom quick tables.

You want the file that starts with the tilde character ~uilding Blocks.dotx. That is supposed to be a temporary locking file.

Your Windows may be hiding these files. Check your Windows View options.

Okay, gotcha, thanks!