blow me by sms backup does not work, did not help either reinstall the application … which can be a problem … backup call functions normally
I have the same problem 
Mine quit backing up at the end of August. Calls and contacts still working. Anyone know why?
Please check if the messages are saved on your Google Drive - if they “only” missing on my avast account it can be a bug on web side, thanks.
I began having the same problem on 11/20/2014. The calls and contacts update as they should. The SMS will not update on one phone. My Note 2 works as it should but my Droid Razr HD will not.
What is your default messaging application?