Everything I've read says there's no hard limit on the number of Content Types you can have in a Library.
However, I'm running into issues with 39 or more Content Types added to a Library. When I add the 39th, the metadata 'details pane' will not show any metadata at all, and it displays a spinning wheel which never goes away, even after leaving it sit all day. It just looks like it's trying to load the metadata, but never does. When I remove the 39th content type bringing the total down to 38, everything works again.
Creating new libraries and adding the content types again. Same issue.
For the 39th content type, I've tried different ones to see if a particular one was causing issue. But it doesn't matter which one is the 39th.
Tried adding content types in smaller batches (five at a time) in case SharePoint had trouble processing all at once.
Tried waiting 24hrs to see if there was a timer job waiting to happen. Same issue.
Here is a screen capture below to help illustrate: