Blender version 2.78rc2, but this is in 2.77 as well as a week-old git master.
OS: Arch Linux.
The same bug is present on an nVidia card with the proprietary 340 driver (it's a GTX 285, and nVidia's 340xx driver is the latest supported driver for the card) as well as Intel integrated graphics. I'm 90% sure this is a Blender bug, anyway.
When using border render in conjunction with real-time viewport rendering and DrawPixels (as opposed to 2D texture or GLSL), the x-y offset of the render border is at the lower left corner of the 3D view.
(You need to have Images Draw Method in settings set to DrawPixels.)
- Set Blender Internal as the renderer.
- Make sure there's something visible in your scene; having a non-transparent world background is enough.
- Set a render border with Ctrl-B; make sure it doesn't reach the lower-left corner.
- Press Shift-Z to enable render preview.
I've also attached a test case file; all you have to do is open it and press Shift-Z.