I think the issue may be related to a conflict between this drag-and-drop functionality of SharePoint for web parts with IE 11's drag-and-drop functionality for images.
Using IE Developer Tools, I put the browser into Edge mode with a web part page in edit mode and tried to drag and drop a web part. When placing the mouse cursor over the web part's header I got the 4-directional arrow cursor, but my attempt to drag
and drop only resulted in selecting text.
With the same page in 10 mode (the default for our SP 2013 environment), I got mixed results. Firstly, when dragging from the header, I saw both a "ghost" image of the entire web part, and also a small blue rectangle with the web part's title.
I believe the "ghost" image was the browser trying to drag-and-drop an image, and the blue rectangle was SharePoint trying to drag and drop the web part, at the same time. When dragging this pair of objects to the intended location for the
web part, on most attempts I would always get the "no" symbol on my mouse cursor at any place on the page and no movement of the web part would occur; on some tries, I would get a cursor like you see in the image below and could successfully
move the web part to a different location; and on some tries, the cursor would mostly display the "no" symbol, but frequently flicker to an arrow for brief instants, and if I released the mouse button at exactly the right moment I might successfully
move the web part.
The image I refer to in the paragraph above is a screenshot I took: https://social.msdn.microsoft.com/Forums/getfile/708378 (Apparently I can't post images until my account is "verified" - but I can't find how to make that happen.)
Putting the browser into 8 mode, drag-and-drop consistently worked fine.
Hopefully this troubleshooting information will be helpful to you.