I have been looking for this movie title for quite a long time. I thought that it was a Danny Kaye movie, but I have looked up all of his movies and I can't match this particular one to any of them.

The movie came out sometime in the 1950s, or maybe possibly in the early 1960s. I don't remember a lot of the movie, just some specific details. It was a comedy heist caper involving an attempted heist in a harem. The treasure room had a small window with two rows of bars across it - one of the members of the heist team was a harem girl who was a contortionist and could work her way through the bars by doing a full 180-degree backward bend of her spine.

The only scene from the movie that I have a clear memory of is near the end. Something has gone wrong and the heist isn't going to work out; all of the characters end up in the treasure room, coming in from different directions - including the harem girl snaking her way through the bars.

Again, I always remembered the lead character being Danny Kaye, but I can't find one of his movies that matches up with this plot. If it wasn't him, then it was another similar actor from the same time.

Any help would be greatly appreciated!

I think you might be looking for The Prince Who Was a Thief... I took the below comment off IMDB:
....Do you remember the part when she flipped in through the window like a gymnast? More like a contortionist. I loved that part. Thats when I fell in love with her :).....
Prince Who Was a Thief 1951 - Tony Curtis - Piper Laurie - YouTube (http://www.youtube.com/watch?v=v4ffNivW-6s)