If you own 2 Xbox 360 consoles, you may be having trouble accessing your purchased XBLA games and DLC on one (or both) of the consoles. Hopefully this information will help get you back in the game.
Marketplace (not Avatar Marketplace) content is given two licenses in order to authorize use.
A static license is given to the console used to originally download the content, allowing everyone using that console to acces it.
A mobile license is given to the profile that purchased the content, allowing that single profile to download and access the content on any other console, while other profiles cannot.
I own two Xbox 360 consoles.
Console A - downstairs
Console B - upstairs
When I make a purchase on Xbox LIVE Marketplace, I sign into Console B with my profile (via Memory Unit) to make the purchase. Then I sign out. This places a "static license" on Console B, allowing anyone to use the content on that console even if I'm not signed into it.
I take my Memory Unit back to Console A and sign into my profile. I then download my newly purchased content onto Console A. Since my profile has a "mobile license" for my purchased content, I can access my content from any console as long as I am signed into Xbox LIVE. But since Console A does not have a "static license" for my purchased content, I must be signed into Xbox LIVE in order to use the content on Console A.
This allows me to share XLBA games and DLC between my two consoles, which comes in very hand when I have friends / family over. They play on Console B, I play on Console A. I can also use Console B to play my games and DLC offline, in the event my internet goes out and I cannot sign into Xbox LIVE. I cannot use my content from Console A if I'm not signed into Xbox LIVE.
If you would like to transfer the static licenses from your Console A to your Console B, you can use the License Migration Tool. The License Migration Tool can only be used once every 4 months.