Ok people, here's a potential temporary fix for you.
This issue seems to be a problem with Chrome's handing of browser cacheing, specifically related to 304 responses from your server, which tell Chrome nothing has changed with a file.
Until Chrome fixes this problem, one thing you can do is turn off browser cacheing, which may slow your site down somewhat. (My server was fast enough not to cause too many issues but this is probably not a good long-term fix).
To turn off browser caching, modify the header.php file in your theme, adding the following adjacent to the other < meta > tags:
<meta http-equiv="cache-control" content="max-age=0" />
<meta http-equiv="cache-control" content="no-cache" />
<meta http-equiv="expires" content="0" />
<meta http-equiv="expires" content="Tue, 01 Jan 1980 1:00:00 GMT" />
<meta http-equiv="pragma" content="no-cache" />
That will instruct the browser not to cache the files, but instead to re-load all files (200 Request) with each refresh. Again, this may slow your site some but at least your users will see the content.
If anyone has any suggestions on making this Chrome-only so that non-Chrome users don't lose cacheing, I am all ears.