Background Images Paths are Changed when CSS Aggregation is On
Thank you @jessehs from Drupal.org.

His solution, which worked for me, is the following:


  Long story short, the problem was that Adaptive Theme appears to be
  doing a "pre-caching caching" of the compiled CSS files in the
  sites/default/files/adaptivetheme/ directory.
  
  The way to get the aggregated files to actually build is to clear all
  caches and immediately save the theme's settings form at
  /admin/appearance/settings/[themename], then do another cache clear,
  just for good measure.


Also included at the source is a helper module that will flush these files when you flush the cache.

Source: https://www.drupal.org/node/1995938