=Сергей=,
Жалоба на проблему в TCP/IP(transport-level error). Может сетевые пакеты режутся гдет по пути?.
Со всеми серверами такое наблюдается?
Из Гугла:( https://www.sqlservercentral.com/Forums/Topic484875-146-1.aspx)
there could be several causes of this bad behaviour

1. network glitches
2. SQL glitches
3. SQL config
4. hostile code

from your description the problem has got worse so it sounds like 1.
- any time TCP connection is broken, your user session will get bombed
- obviously will not apply to connection from local console so try backups from there

I am not aware of other standard settings that deliberately cause idle SPIDs to be KILL'ed
- perhaps some GPO or 3rd-party tools might do such (and virusses could do similar)

certainly the gemlins will leave footprints in the Windows EventLog and/or SQL logs
- you should be able to diagnose from there