Sounds like the local profile is out of date or corrupted, and when an app or routine asks for data from it, it causes the involuntary disconnection. I'd start by clearing the system cache (it's just temporary files plus game updates, automatically reloaded) by:
1) Tap the silver Xbox Guide button on the controller, go 2 clicks right to Settings, pick System Settings.
2) Select Storage (or Memory if Storage).
3) Move over any hard drive or memory unit (not the Cloud) & press (Y) on your controller & pick Clear System Cache.
4) When asked to confirm storage device maintenance, choose Yes.
Delete profile from your console
1) go to http://xbox.com/accounts & click on Sign in at upper right - if you can't sign in then ** STOP: DON'T ** delete your profile. Please call Xbox Support http://tinyurl.com/XBLnumbers Option 2 for help with your account. If you can sign in, delete & redownload your profile on your Xbox this way:
2) Go to Settings - pick System - choose Storage - pick All Devices - choose Gamer Profiles
3) pick the gamertag choose Delete, and pick Delete Profile Only.
This deletes the profile but leaves saved games and achievements. Now, re-download your profile.
1) Tap the silver Xbox Guide button
2) Select Download Profile (and if you don't see it, press the blue X button to make sure you're not signed in) and proceed with Profile Download.
3) Enter the Microsoft Account addresse for your Xbox LIVE gamertag when asked.
4) Enter the password for the Microsoft Account.
4A) If your account is an Xbox LIVE child account, the parent’s Windows Live ID & password is required as well to download your profile.
5) Pick the storage device for your profile (choose Hard Drive if you have one, or Memory Unit if you don't have a Hard Drive).
6) When the download completes you will be asked if you want the console to remember your password. If it's your home console, I'd say yes, otherwise I'd think carefully about leaving my profile unprotected on somebody else's Xbox!
7) Sign in, and retry.