the first thing I would suggest is to clear the browser caches and "Force-refresh" the page in your browser.
But probably you have already done that, so ...
THe next thing I would try is to go to one of the computers that is misbehaving and find out what URL it is trying to access for the background image.
- If the image is within one of the HTML tags, then you can use the browser's "View source" menu option to open the source HTML and find the URL.
- If the image is defined in a stylesheet (i.e. a ".css" file), then you will need to get the URL of the css file (again, using the browers's "View source" menu option), and then paste the stylesheet's URL into the browser's address bar.
Once you have the URL of the background, check it is an absolute URL, then paste it into the address bar of your browser. If it doesn't show, then go to the course files folder of your Moodle site (or whereever the image is stored), navigate to the image file and click on it. The browser should display the image.
That's enough for now. Please let me know what happens, and we can take it from there.