This happens with Firefox and possibly other browsers as well (only tested in Safari and Firefox so far). I have a wrapper div (100% width) that has a centered background image that is around 1400px wide. There are a couple of pixel precise elements in this background image - when the Firefox browser window is an odd width (say 1007 pixels wide) it causes the background image to shift 1px to the right, breaking where the background and other elements line-up.
Yes, always provide a link.
It's hard to imagine how else this could be handled. You can't have half pixels, so the user agent has to round up or round down.
So, you have to work round this, generally by wrapping all your pixel based alignment essential items in a wrapper that has a fixed width, without seeing your design it's hard to be more specific, but I'm sure there's a workaround.