Not sure I understand the question. A background to what? The body? The main content area? Something else?
If it’s the body, you can never guarantee it won’t ever have to tile, short of specifying no-repeat (or repeat-x or repeat-y as appropriate) in your css of course, because you can’t account for every possible viewport size that exists or will ever exist.
If you’re putting a background into a region with a fixed width, you can make your image that width and so it won’t tile along the x-axis. But heights are usually determined by the content, so you’ll still need to tile along the y-axis or specify background-repeat: repeat-x
to suppress vertical tiling.
The only way you can guarantee to have an image fit is if you know in advance the size of the width and height of the region. This is very rare in Web pages.
And remember that large images take a long time to load, I suppose I should say.
(Usual advice about using a child theme and not altering the parent – if you’re not already doing so. Using a child theme is essential for any of the default themes, like twentyeleven, which you should never alter.)
Cheers
PAE