Is it possible to combine a Bootstrap theme with an existing custom Drupal 6 theme?
There is the Bootstrap base theme for D7 (and even an upcoming D8 version), which bridges the gap between Drupal and the Bootstrap Framework, with also quite a few (contributed) sub-themes available for it already.

So if your question was about D7, then it seems such combination that you're looking for should be possible. As you may know already, the Bootstrap base is not available for D6. But maybe you can use the D7-version of it as inspiration to somehow make such combination also work for D6. Ultimately even come up with a backport of the Bootstrap base theme for D6?

Note: as mentioned by Molot's in the comment to this answer, there is "some" D6 backport of it available on github ... Via Drupal.org there doesn't seem to be any D6 versions, however there "is" some D5 version of it also ... Maybe upgrading the D5 version to D6 is another approach to consider? Using Browse code repository only results in D7/D8 versions though ...