If you have removed system apps or applied framework changes (like a mod) then the OTA could fail cuz of that. It could even be from your custom recovery. There should be an error code that gets saved in /cache or it should give you a reason why it fails as it is attempting to apply the update. Have you done any permanent changes similar to what I outlined?
If you have no issues with wiping the entire phone, then you could use Odin (Windows only program) to apply the 4.3 update. Now, I must say that you will not have any option to use AOSP roms or custom kernels once on 4.3. If this matters to you, then I can help you get rid of the ota notification.
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[VZW] Samsung Galaxy SIII