I stumbled upon the answer, but I don't understand all the details. I'll
investigate further, but here's a summary:

I'm running server 3.1.2.20, backup client 3.1.06, and SQL connect agent
1.1.1. My client node is called "ntserver50", and was already being backed
up by the server every day. When I installed the connect agent, the
configuration panel suggested a node name of "sql_ntserver50". I had no
reason to doubt that was a good name, and I guessed that I'd need a
different node name for my SQL server backups anyhow, since was planning to
assign the new node to a different policy domain.

The ADSM server is absolutely silent about any error message of any kind
that I could find. The connection was simply dropping, and the backup
failed. The fix was to simply change the SQL connect client node name to
"ntserver50", instead of "sql_ntserver50". The SQL backups work just fine.

I haven't had time yet to investigate the impact of this having two node
names mapped to two different types of backups (filespace and database).
Maybe someone experienced can guide me.

Feel free to ask further questions and/or make (intelligent) suggestions.
I'll post an update after I look into this further. Can anyone point me to
documentation that I should have read, that would have prevented this
problem?  If the server had given me any reasonable error message, I could
have resolved this quickly, I think.

Thanks for your input.