There is another thread on this topic. Verizon has apparently tagged it as "answered," but it has not been fully answered.
The other thread is backup assistant plus lost network connection(36). please try again.
I agree with the original poster in the other thread that this problem started about the time of the last system update, 08/21/15. However, besides Backup Assistant failing to sync, I've discovered that once Backup Assistant attempts to sync, I lose all internet access through WiFi connection, even though my phone shows a strong WiFi signal. I am only able to regain internet access through WiFi if I "reset settings." The downside to this is that I have to reset my wallpaper, ring tones, font size, etc..... and that only lasts a day until Backup Assistant tries to run again.
I have not found any way to disable or remove Backup Assistant on my phone (Samsung Galaxy S4). Even if I switch to Verizon Cloud as suggested by VZW representative EbonyR_VZW, how does this solve my problem with Backup Assistant? Backup Assistant will continue to attempt to operate once every day, meaning I would then also have to "reset settings" every day. Backup Assistant is listed as an "Account" on my phone, but I've also found no way to remove this feature by accessing my account online.
This issue needs further guidance from VZW.