I posted an answer on another question on another topic that might have some use here. That answer involved SMB connections, not SQL. However it was identical in that it involved a low-level transport error.
What we found was that in a heavy load situation, it was fairly easy for the remote server to time out connections at the TCP layer simply because the server was busy. Part of the reason was the defaults for how many times TCP will retransmit data on Windows weren’t appropriate for our situation.
Take a look at the registry settings for tuning TCP/IP on Windows. In particular you want to look at TcpMaxDataRetransmissions and maybe TcpMaxConnectRetransmissions. These default to 5 and 2 respectively, try upping them a little bit on the client system and duplicate the load situation.
Don’t go crazy! TCP doubles the timeout with each successive retransmission, so the timeout behavior for bad connections can go exponential on you if you increase these too much. As I recall upping TcpMaxDataRetransmissions to 6 or 7 solved our problem in the vast majority of cases.