com.apple.mobilephone error 1035 meaning?

I had this problem, and my voicemail box was full, I deleted several voicemails and then it worked. 

Problem solved:

Go into your VM by pressing and holding one, go into personal options, administrative options, then turn off password. Done

My telephone now receives calls but does not correctly display new Voice Mail messages.  Interestingly, it displays the little red circle on the green phone icon, but not on the VM icon.  When I try to retrieve VM using the VM icon at the bottom of the screen, I am  unable to do so.  The red balloon will not disappear until I access my VM via  the press 1 method to access VM. 

I have rebooted the phone.  I now have removed the password protection from the VM as suggested by markrjinlv above to see if this is a practicle workaround.  Regardless, even if it does, it STILL means that the VM interface of iphone with AT&T is faulty.  

Thank you! 

The voice mail prompts worked. I had reset networks, turned off WiFi, visited the AT&T store where they gave up and referred me to Apple. (They put it on a diagnositic machine it showed me that it said it had a software problem.

I spoke to 2  Apple support people, one a 'supervisor' who said I needed hands on with an Apple genius at the store. I went to the Apple store, willing to wait - was told first appointment was Thursday - and this was Saturday, and this is a business phone. An employee had some sympathy and played with it a bit - and said anything with voicemail is a carrier problem, and I should go back there. 

What astounds me is that none of these people working for AT&T or Apple suggested this solution.

Whew! very glad you posted!

Thanks! Suggestion 29 worked for me on my i7. WiFi off Data on Cloud drive off. Recorded my vm message and it took! Thanks again!