To start with, never have more than one font manager on any system at the same time. They will fight each other over control of which fonts are enabled/disabled. Pick one and completely remove any others. That includes removing Font Book if your preferred font manager is Suitcase, FontExplorer X Pro, or other app.
From there, you have to go down the line of possibilities. From easiest to test and fix, to the most time consuming:
1) Corrupt fonts will damage Font Book's database. And rather easily at that. You can clear it, have the fonts working for a day or two, and then the damaged fonts toast the database again and you're back to a non functioning Font Book. To completely reset Font Book:
Restart your Mac and immediately hold down the Shift key when you hear the startup chime to boot into Safe Mode. Keep holding the Shift key until you see a progress bar towards the bottom of the screen. You can let go of the Shift key at that point.
OS X asks you to log in (you will get this screen on a Safe Mode boot even if your Mac is set to automatically log in). Let the Mac finish booting to the desktop and then restart normally. This will clear Font Book's database and the cache files of the user account you logged into in Safe Mode.
When you launch Font Book after booting normally, a new database will be built. If the problems return, you likely have bad fonts on your system somewhere.
2) So the next step is to get the original fonts back on the system, and only the original OS X supplied fonts. Move all fonts out of your user account Fonts folder to a temporary folder on the desktop. Move all fonts out of the root /Library/Fonts/ folder to another temporary folder. Download Pacifist. Follow the instructions at the bottom of Font Management in OS X to reinstall just the fonts OS X supplies.
For Mavericks, that would require first downloading the full installer .dmg file through the App Store. When the installation procedure starts after the download is complete, stop it by pressing Command+Q. Note: the installer is supposed to remain on the drive in the Applications folder when you quit the installation, but it sometimes gets deleted anyway and you have to download it all over again. So before you quit the app, just leave the initial installation screen up, go back to the desktop and copy the installer somewhere else, then quit. If you end up with both, then just delete one or the other. Now use Pacifist to reinstall OS X's supplied fonts, it will overwrite all of the fonts on the drive will all new, fresh copies.
Repeat the steps above to clear Font Book's database and font cache files.
If, with only freshly installed OS X supplied font on the system you still have Font Book randomly turning fonts on or off, then the OS needs to be reinstalled.
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