Drupal 6 + Twitter Bootstrap 3 theme
Prerit,

I think there is indeed one custom port to BS3 and that can be used in Drupal 6. Follow post #3 to get the idea. However, this is a custom theme made by an Drupal developer and its not in the drupal's project yet.
FYI, it does support jQuery version 1.7.2 by un-setting the default jquery 1.2.6 version which intern may break some functionality in your site.
The link to the Github Repo.

Personally I would suggest to use Zen or Omega for a Drupal 6.x site.