Context: I am using a software package called Social Engine. It is extremely buggy.
Anyway, I asked (paid even) the Social Engine people to do an upgrade and when
The issue could be due to your browser having cached the original CSS files and trying to request byte ranges of the new files. See for example this bug with Chrome.
As far as disabling range requesting, it might not be necessary if you clear your browser cache, but if you need it you could try the following Apache config:
Header unset Accept-Ranges
This will tell clients that they cannot use the Range request header to request byte ranges of your files.