Wow. I may have seen that show or at least some of it.
edit: If it was on HBO I guess I did not see it. I may have thought of something on the History or Discovery Channel.
If I recall the Saturn V had 3 stages. The big booster, the smaller 2nd stage and the command,service module with the LEM in between 2nd and upper stage.
I had to think about it for a moment.
edit:
This vehicle had 3 stages. The 3rd stage had CSM and LEM, etc. The 2nd was the same diameter as the first.