We're currently running ArcGIS 10.4.1 and SQL Server 2012. I needed to do some testing so I created a new user in SQL Server Management Studio with a database login and assigned db_datareader access. We have 4 databases running in this instance, and I successfully connected to 3 of the databases. However, I received the following error on the 4th "Failed to connect to the specified server. Do you want to continue? Operation Failed". The only thing that was noticeably different was that in the failed database db_datareader did not have an owned schema (whereas the other 3 both had "db_datareader" checked). I checked it and I received the same error message.
Let me know if I can provide any additional info. Any help would be greatly appreciated!