If you do let your Windows Live ID (such as a Hotmail address) but still have a valid gamertag, the lapsing of the former will also invalidate a perfectly valid one of the latter.
If this happens then, according to Microsoft, "The only alternative is to set up a new Gamertag with your preferred email address."
Although this affects a small number of users, Microsoft has owned up to it, saying, "We are investigating this issue and don't have any further comment to make right now. Any consumer with any issues regarding their Xbox 360 or Xbox Live account should visit xbox.com/support or call their country's customer support number."Source: GamesIndustry.