Sql-server – The database theDb is not accessible. (ObjectExplorer)

Most likely it's a permissions problem on the new server. Your SQL logins don't carry over automatically and if you manually create it on the new server it'll get a different SID which won't jive with the one that's granted access to the database.

Best way is to use sp_help_revlogin (KB918992 in case the link ever dies) to generate a script of the logins on the old server and then run the script on the new server. This procedure will carry over SID and password information to the new server effectively recreating the login on the new server. Once this is done the permissions set up in your database should then be in sync with the login(s) on the new server.

This shouldn't be a problem for Windows logins as SQL gets the SID from Windows, although you'll still need to set up the Windows login on the new server. sp_help_revlogin will handle this part for you as well.

Are you running VSS backups on the server (NTbackup, BE, etc.) for stuff other than the SQL data? According to this, the SQL differential backups can be invalidated if a backup software uses VSS for snapshots.

As far as I know, LS does not use VSS just like Nick said. According to this thread the fix was to disable the SQL Writer Service.

Update based on comments:

Because you are using NTbackup to backup other files on the server, the fix would be to disable the SQL Writer Service.

When NTbackup runs it calls ALL the available VSS writers, even if you don't need them. This means that the SQL writer will put the DB into a consistent state (flush cache to disk) while the snapshot is created. The DB then thinks it has been backed up, even if you didn't copy the information from the shadow copy. This is what's breaking your backups.

One thing you can check to see if VSS is running against the DBs (which I'm positive it is and breaking the chain), is look in the SQL log (I think its in the error log). You should see freeze/thaw events during the time your NTbackup jobs run.