My first Xbox 360 was a family console, and after its optical drive broke, we got a new Arcade and popped the old hard drive in it. Everything worked fine, except for one thing: any game content we had paid for and downloaded from the Marketplace wouldn't work unless the owner of said content was signed in. Being the oldest child of the family, and the only one with an Xbox Live Gold account, this meant that my account had to be logged in at all times to play Castle Crashers, or to play the Metallica songs on Rock Band, etc. 
 
Anyways, now I'm in college, and I have my own 360. My roommate wanted to play some Rez HD, but it wouldn't let him play past the content that's in the trial version. 
 
After some investigation, it turns out that Xbox 360 content is tied to both your Gamertag and your console. You're only allowed to re-assign the console ID associated with your paid-for content once a year, and you have no granular control over which items go to which console; you have to mass-assign them to one console. Here's the link if anyone wants to do this; it was mildly difficult to find.
 
That's not the worst of it, though: once you've gone through that process, you have to re-download everything on your new console, even if you already have it there. So I had to go through and delete most of my Arcade games and queue up like ten downloads for games that I already had on my hard drive! 
 
So that kind of sucks.