Object Mode is in constant X-Ray mode? A bug?

The problem is that while in Object Mode it is behaved like it is in X-Ray mode all the time, even if the mode is switched off and the objects look solid.
I can select through objects, can’t deselect properly, because there are always some other objects getting selected instead, Box Selection of multiple objects is a nightmare and unusable.
Actually it’s pretty much annoying and destructive thing.

I hope to get some info how can I avoid this bug and if there is an add-on or some other solution.
May be it is a well known bug to be fixed in the nearest future?
Thanks in advance.

Windows 10 x64.