So, I don't get it. Why does this text block work for some lines, but not others? Is there something specific I have to do with the line on the right?

Hi,

Based on my test, if we double-click the line shape and then add the text string, the text will not overlap the line shape. If we use a text box, the line shape is across the text. Please see the image below:

Thus, do you mind sharing us a sample file (Upload to OneDrive or email: ibsofc@microsoft.com add the thread URL as the mail subject)? We want to test it.

Regards,

George Zhao
TechNet Community Support