Text refuses to follow certain path

the curved path should work just like the angled one does. Can you share the file with the uncooperative line (via dropbox, or some other file sending service)? This would be easier to diagnose with the problem file in hand. 

Presumably you've tried the Type on a Path tool (which is underneath the Type tool).

How were those lines made? I cannot get text to flow onto either of them, but if I draw an identical-looking line in your document with the pen tool, it works just fine.

I'm not sure what you did to those paths, but I was able to use them by first selecting them and going to Object > Content > and changing from text to Unassigned, at which point they show in the layers panel as Compound paths, so I went to Object > Path > Release Compound path, and they work, but they seem to be made from two paths each which are not quite aligned. You can fix that by using the Direct Select tool to marquee around the end points that should be together, then Object > Paths > Join to make a single path

I also apparently have the same problem. Using InDesign CC, I can create a path with the pen tool. I can edit it with the direct selection tool. But when I switch to the text-on-a-path tool I can wait forever for that plus sign. It never shows up. When I just position over the path and start typing, I get straight inline horizontal  text. The layers panel shows it as a simple path, nothing weird about it. Is the text-on-a-path tool broken?

Hi Joe, I just tested the Pen tool with the Text-on-Path tool. Nothing unusual.
InDesign CC 2015.3 on Mac OSX 10.10.3 German.

What is your exact version of InDesign and OS?

Uwe

I am using InDesign 2015.3 Release  11.3.0.34 x64 Build, running on Windows7 on a Dell desktop.

I am using InDesign 2015.3 Release  11.3.0.34 x64 Build, running on Windows7 on a Dell desktop. US English.