Both IE and Chrome, IE is a version 10.
Did not yet try to drag and drop documents in a list, just to see if that will provoke a error message, like Hnery suggested.
Ensure you are using 32-Bit IE, which it definitely should be if it is IE 11 as I think there is no 64-Bit IE 11 version.
You could also try explorer view? I find in SharePoint that there are many ways of doing the same thing and sometimes one way gives you not much in the way of an error but others do but from all that you have said so far I am sure it is client side.
As someone else has commented it would be worth trying a different browser.
It is also an issue on the Server, and it also touches the upload of documents to SP.

