To continue processing after an error you must turn off XACT_ABORT then check and respond to (or ignore) the error yourself.
DECLARE @iErrorID INT
SET XACT_ABORT OFF
INSERT INTO TABLE
SELECT @iErrorID = @@ERROR
THEN use IF logic to respond to error number according to your desires.
Additionally, you will still get error raised if you has an invalid column or table name or similar type of error.
Glenn