I recently tried the new Blender 2.80 version which (apart from other things) also changes the internal file structure and the FBX export. Because of this, the behind-the-scenes FBX import in Unity doesn't work anymore. “Blender could not convert the .blend file to FBX file, […] you need to use 2.58 or later versions.” I didn't find any other threads mentioning this, so I'm posting this in case of other people running into the same problem. It's not really a bug since the new Blender version is still in beta (and according to estimations, won't be released until spring, so hopefully the Unity team will have enough time to provide a fix). The solution, for now, is to either keep using the old version 2.79 – or to export the FBX file manually. Files saved with version 2.80 cannot be opened in earlier versions (the only way is to export from the beta file and import in the earlier version, which causes e.g. modifiers to be lost).