You get two licences when you purchase content on the Xbox Marketplace.
A console licence - You can play the content on any profile on the original console that the content was downloaded on.
An Live account licence - You can play the content on any console provided that the Live account that purchased the content is signed into Xbox Live.
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If you think some of the games are attached to another console you can tranfer them here: http://www.xbox.com/en-GB/Support/LicenseMigration/Home