connection was attempted with an older version of SQL Server client communications software that is not supported
Accepted Solutions
NOTE: Even though Derek Law 's link doesn't list/include 10.4.1, I just install the
Microsoft_SQL_Server_2012_SP3_Native_Client_64bit_149470 version of the client on my test 10.4.1 machine, and I was able to connect to what I believe is a SQL 2008 (SP4) SDE database....other than drawing something quickly, I did not test any further.
...this is critical for me before I upgrade too. 
Derek and Rebecca,
When reviewing the technical requirements for ArcGIS Desktop 10.4.x I noticed that SQL Server 2008 was not on the list of supported SQL platforms - Rebecca pointed this out as well. Thank you to Derek for providing the KB article to verify this information. I installed SQL Server 2012 Native Client on a test machine and was able to add a connection to a SQL Server 2008 R2 database.
Hopefully this will help someone else in a similar situation. Our organization is currently in the process of modernizing many of our enterprise applications to the most current version possible. SQL Server is likely next as we migrate away from SQL Server 2008 R2.
