Now that we have all the explanations as to why this is happening. Let's see an actual solution to the problem.
First let's take the statements that @d-k-mulligan proposed above and turn them into stored procs.
IF OBJECT_ID('dbo.prcIDontExistINSERT', 'P') IS NOT NULL DROP PROCEDURE dbo.prcIDontExistINSERT
GO
CREATE PROCEDURE dbo.prcIDontExistINSERT
AS
BEGIN TRY
INSERT INTO IDontExist(ProductID)
VALUES(1)
END TRY
BEGIN CATCH
SELECT 'There was an error! ' + ERROR_MESSAGE()
END CATCH
GO
IF OBJECT_ID('dbo.prcIDontExistSELECT', 'P') IS NOT NULL DROP PROCEDURE dbo.prcIDontExistSELECT
GO
CREATE PROCEDURE dbo.prcIDontExistSELECT
AS
BEGIN TRY
SELECT * FROM IDontExist
END TRY
BEGIN CATCH
SELECT 'There was an error! ' + ERROR_MESSAGE()
END CATCH
GO
If we run either of them we see the same error.
EXEC dbo.prcIDontExistINSERT
EXEC dbo.prcIDontExistSELECT
Msg 208, Level 16, State 1, Procedure prcIDontExistSELECT, Line 4
Invalid object name 'IDontExist'.
The solution now is to create error handling wrapper procs with the sole purpose of catching any error from the original procs above that are getting the object not found errors.
IF OBJECT_ID('dbo.prcIDontExistInsert_ERROR_HANDLER', 'P') IS NOT NULL DROP PROCEDURE dbo.prcIDontExistInsert_ERROR_HANDLER
GO
CREATE PROCEDURE dbo.prcIDontExistInsert_ERROR_HANDLER
AS
BEGIN TRY
EXEC dbo.prcIDontExistINSERT
END TRY
BEGIN CATCH
SELECT 'There was an error! ' + ERROR_MESSAGE()
END CATCH
GO
IF OBJECT_ID('dbo.prcIDontExistSELECT_ERROR_HANDLER', 'P') IS NOT NULL DROP PROCEDURE dbo.prcIDontExistSELECT_ERROR_HANDLER
GO
CREATE PROCEDURE dbo.prcIDontExistSELECT_ERROR_HANDLER
AS
BEGIN TRY
EXEC dbo.prcIDontExistSELECT
END TRY
BEGIN CATCH
SELECT 'There was an error! ' + ERROR_MESSAGE()
END CATCH
GO
Finally, let's run either of our error handling procs and see the message we expect.
EXEC dbo.prcIDontExistInsert_ERROR_HANDLER
EXEC dbo.prcIDontExistSELECT_ERROR_HANDLER
There was an error! Invalid object name 'IDontExist'.
NOTE: Kalman Toth did all the hard research work here:
http://www.sqlusa.com/articles2008/trycatch/