I want to start supporting page widths below 960px (non-responsive), but I know this is not traditional.
When I say non-responsive, I only mean that the page layout not change dramatically like you'd see on a phone. I would utilize min-width on the container to go below 960x, with a width on the container having the ability to grow to 1000 or so.

The reason I want to do so is for windows 8 splitscreen docking without scrollbars. If you open some of the most popular websites and dock them on an 1920 monitor you'll see that they have scrollbars. Even the MSN site uses 960 width and would have scrollbars when docked if the content didn't resize when scrunched.

960px is half of 1920 yes, but we've got the browser frame and vertical scrollbar to consider as well. Which on a 1920 screen resolution in windows 8 with chrome is 927px. So if you fit your page into a 927px container, people with large screen resolution can splitscreen dock without having a horizontal scrollbar.

I was even considering supporting lower screen sizes than that due to the extremely common 1440 lenovo desktop monitors and 13-inch macbook air resolutions to splitscreen without horizontal scrollbars, which would mean fitting the page content within a 687px container. (assuming that the vertical scrollbar + browser frame consumes 33px or less)

So anyways to the question: Should I support minimum page widths below 960px?