If the machine name was changed after the SQL server have been installed some connnections will fail. Check these :

sp_helpserver 'SERVERNAME'

Select SERVERPROPERTY ('SERVERNAME')

SELECT SERVERPROPERTY ('SERVERNAME')

SELECT @@SERVERNAME

You cannot catch that error from inside the SP because its execution will never reach IF @@ERROR ...

But you can catch that error from the calling stored procedure or connection.

DECLARE