This is about Twitter Bootstrap 2.x. Good children should use 3.x.
The behavior of Twitter Bootstrap 2.x modals was strange in IE11. Other browsers were fine, but only IE11 had problems…
To just write the result: updating the version from 2.0.4 to 2.3.2 (the latest version of 2.x) fixed it.
Since encountering bugs like this makes you sad, I want to keep the libraries I’m using regularly updated.
That’s all from the Gemba.