I just realized I am having the exact same issue as Rob and Allen, within the last week upgraded desktops to 10.6.1 from 10.6.0, and we are not using SQL Express.
When trying to export a feature class from the SQL geodatabase in ArcMAP (right-click layer): Error exporting data. "Open featureclass failed". Then: "Warning: Some records may not have been exported correctly". "There was an error exporting the layer".
Same as Rob notes,.. Selecting ALL records in the feature class and then exporting works fine, as does using ArcCatalog to export to a file geodatabase.
Geodatabase is 10.6.0 on SQL Server 2016, which is less than a year old.
Very strange, have not had problems previously.
Brad