Okay, this is making a lot more sense now -- thanks so much for the screenshot!
The game is actually running at a larger resolution, but your monitor is just cropping rather than scaling the image. If you use a smaller resolution that should work, but I imagine that if any other DirectX/OpenGL programs (or at least Unity 3D programs for sure) are run at this larger resolution, then that would cause the problem. It may be something funky with your graphics card drivers (you could try updating them), or some limitation with your monitor (you could look that up, or there may even be updated drivers for them). I once had an older laptop that did something similar to this, but in reverse: if I ran programs fullscreen at smaller resolutions than its 800x600 default, then it showed them with black borders, rather than scaling the image up like you'd expect.
At any rate, regarding why there are black bars at that resolution, I'd point you to this: http://arcengames.com/mediawiki/index.php?title=Tidalis_-_Common_Technical_Support_Questions#I_set_the_screen_resolution_higher_than_1280x768.2C_and_now_I_have_black_bars
From the look of things, you'll want to use a smaller resolution (if the driver update doesn't work), because it isn't scaling down like it should to fit your actual monitor pixels (apparently 1680x1050 is more pixels than is native to your display, I'd guess), or this is some ultra rare bug in Unity 3D, or this is some problem with your driver.
Thanks much for the added clarity, this is definitely a new one on me! Sorry for the trouble you're having there.