It appears to be a very popular issue that has many causes and solutions.
Here is my 2 cents with a solution that worked for me...
First, the symptoms of my situation:
Calendar sync stopped working, everything else still gets sync'ed
- The calendar stopped synchronizing any new entry and just showed the cached existing items up to a recent date.
- Any new item added on the phone would not be saved. Any new item added from a PC would not be sync'ed into the phone.
- "Calendar" had disappeared from the Sync list in the Android settings, all the other apps were still listed and working.
- There was no calendar listed under my account in the Calendar settings
- Trying to add a calendar would bring a prompt for Sign-in a New or Existing Account which would consistently fail with the "couldn't sign in" error.
My working resolution (root needed):
- Removed the calendar.apk from system/app folder
- Restarted the phone
- From Play, located the Calendar app from Google and installed it.
From this moment, the Calendar should be back in the Sync list
- (Optional but recommended) moved the new com.google.android.calendar.apk from /data/app to /system/app and restarted the phone. This turns the new Calendar into a system app.
This replaces the stock/AOSP calendar.apk with the much bigger (20MB vs 2.6MB) but nicer one from the Play Store. In my case, this was a necessary part of the solution because, doing the same procedure as above with this stock apk did not help at all.
One problem, many possible causes and solutions:
I can only speculate on the causes as this happened after no obvious configuration change or application installation. I suspect the stock calendar.apk (4.4.4-1227136) in my official Android 4.4.4 from THL to have some kind of bug.
Other people have resolved a similar problem by either updating/re-installing Google Play service,clearing the cache and data for Google Play Framework, disabling the 2-step authentication (not used in my case), removing the Google account and re-adding it, replacing the Calendar Sync Adapter apk, ...