Can I use Twitter Bootstrap and jQuery UI at the same time?

Check out jquery-ui-bootstrap. From the README:

Twitter's Bootstrap was one of my favorite projects to come out of 2011, but having used it regularly it left me wanting two things:

The ability to work side-by-side with jQuery UI (something which caused a number of widgets to break visually) The ability to theme jQuery UI widgets using Bootstrap styles. Whilst I love jQuery UI, I (like others) find some of the current themes to look a little dated. My hope is that this theme provides a decent alternative for others that feel the same. To clarify, this project doesn't aim or intend to replace Twitter Bootstrap. It merely provides a jQuery UI-compatible theme inspired by Bootstrap's design. It also provides a version of Bootstrap CSS with a few (minor) sections commented out which enable the theme to work along-side it.

just to update this, bootstrap v2 no longer conflicts with jquery ui

https://github.com/twbs/bootstrap/issues/171

Edit: as @Freshblood there are a few things that still conflict. However, as originally posted Twitter suggests that they are working on this and it largely works, specially compared to v1.