In case what you're seeing on my site now doesn't look like that, I've included a screenshot. By default, the Drupal Bootstrap
3 theme cannot do this. To me, that seems odd because it's one of the more 'defining' visual features of a typical bootstrapped web page. Even the Twitter Bootstrap page
itself does this currently. Now, if you so desired you could easily put in the content-width version of this out of the box. See below.
But even if you put in the recommended markup to achieve the full-width version, you are still stuck inside that block container (which is 'Top Bar', by the way... throw your jumbotron in that block region to get something like above!). The problem is that there is no block region outside of the main container (which is set to a static width). I have a solution! It isn't 'turn-key' (you can't set the background image via the GUI), but it works. Now you have 3 options... you can download my entire theme, you can download the files below and replace them in your theme, or you can scroll down a bit and see the code that I changed.