Is it possible to lose badges after they've been awarded?
No, badges are never taken away. That's part of the site's design.
However, if you no longer meet the criteria for a badge (say your "great question" got busted back down to 99 votes), then you won't earn another one of those badges until you "catch up."
General badges will not be taken away once you have earned them (barring very exceptional circumstances).
Tag badges, on the other hand, will be revoked if you don't meet the criteria for them anymore. If you earned a bronze badge in a certain tag for 20 non-wiki answers with a total score of exactly 100, and then one of your answers gets downvoted (bringing your total score to 99), you will lose the tag badge.
No, once they are awarded they are never taken away.