How to fix a field with unlimited values where the "Add another item" is broken? I am working on Drupal 7.43.

I have created a few months ago standard fields (text) with unlimited values. Everything was fine.

Today, I can not add another item, the Javascript icon is turning and then nothing happens. Surprisingly, it works on a local version. The weirdest thing is when I use a fresh field, created for test, everything works fine and I can have multiple values added.

In recent log messages, I have an AJAX warning : Invalid form POST data. (EDIT : this warning message is not displayed each time I try to add another item.)

I have tried many different things (jQuery Update configuration, new theme, deactivation/reactivation of modules, cache flushing...).

Does anyone know if I could fix my "broken fields" so I can still use them instead of creating new ones and cloning all the values already entered?
Thanks in advance.

A similar issue here

EDIT (05/09/16) :

Looking at the Firebug Console, in JSON tab, I have just noticed that 2 lines are missing, compared to my localhost version, where the function works fine. The lines missing : 


Object {command="add_css"...}
Object {command="insert", method="prepend", selector="head"...}.


Seems important? Any way to get these lines on the production website?
After many different tests, I have found the solution.

All the multi-value core fields using the module "Better Field Descriptions" were broken.

The help text added below by this module seemed to block the new option line to be created or at least displayed. Deactivate the module fixed the problem.

So, if you need a multiple values field and a permission to edit the help texts, be cautious with module interactions.

Thanks for those who tried to help me on this issue.