The next step requires extremely careful planning.
I had an equally serious (but different) problem. This is what I did to resolve the issue. First I did a simple Reset (Home and on/off until the Apple logo appears) and it didn't help. Then I did a Restore with a reload from the backup and that also accomplished nothing. Then, I did a Restore as new with NO reload from backup and that resolved my issue.
IF you choose to try the same thing, here are some cautions. Make absolutely certain that all purchased data, such as apps, music, movies, etc., are present in the iTunes library. Also make absolutely certain that all non-purchased data, such as photos, pdf files, Word files, etc., are present and accessible somewhere on the computer's hard drive. Be aware that you will probably lose all game data (scores, etc.). I did not investigate a solution for that because I don't play many games and I really don't care about the game history. Be forewarned that it is a bit tedius recreating all your folders and putting each app where you want it. Finally, be aware that most (all?) passwords will be wiped out and most (all?) settings will have returned to their default values.