I've run into this before, and it wasn't fun to fix.
Go into settings, accounts, and open the affected Google account. Check to see when the last sync was between Google and your phone's contacts. If sync is off, you've found the issue. Turn it back on- Issue fixed. If it hasn't synced in a long time but is still on, continue to the next paragraph. If it just synced, skip the next paragraph and go to the following one.
Turn off the contacts synchronization in the affected account. Before proceeding, make sure that the web version of Google contacts displays everything perfectly and is not missing any contacts, because your phone's data is about to be deleted. Update the online versions if necessary. Then go into the apps list (settings-apps) and you will find one called Contacts Storage. While this isn't an app you can open per say, it contains all of the contacts in the phone, and is the source of the issue. Clear the cache and data of this app, then turn contact synchronization on the Google account back on. After a few minutes, all of your contacts will come back and the data should be correct. Comment on this answer if that does not work, because I ended up having to do more steps than just this, even though I shouldn't have.
This paragraph is if the data on synchronization says it is updated. Open the Google contacts online version and the contact on your phone and do a comparison. If they are the same, then everything is working properly and there is an issue in your contacts with Google, not the phone. If they are different, there is some witchcraft going on that even I can't explain. If the data on the web version of Google is correct but the phone isn't, follow the procedure in the preceeding paragraph. If the phone is correct but Google isn't, you're going to have to manually fix the online version and then follow the procedure on the preceeding paragraph.
I apologize for the large number of cases in this answer, but there are many different ways that this could be addressed this and I wanted to cover all of them.
