Problem: There are issues with the Alter Field tool not immediately showing the results or throwing inconsistent errors depending on whether the user is the owner of the SDE feature class.
Conclusion: Using the Alter Field tool in SDE can have different results depending on ownership. The same operations performed on an identical schema in a file geodatabase did not produce any errors.
Solution: Depending on ownership, when the tool doesn't block you from running it, exit ArcCatalog and come back in. The results of the Alter Field changes will be shown in the Fields tab and the error on checking the Properties will not occur.
Database: SQL Server Enterprise Database (SDE un-versioned)
Application: ArcCatalog 10.3.1 (version 10.3.1.4959)