Here's an example of the scenario:
CREATE TABLE dbo.TestConversionTable
(
    Value VARCHAR(MAX)
)
INSERT INTO dbo.TestConversionTable VALUES ('1')
INSERT INTO dbo.TestConversionTable VALUES ('foo')
CREATE VIEW dbo.TestConversion 
AS
    SELECT CONVERT(BIGINT,Value) Value
    FROM dbo.TestConversionTable
    WHERE Value <> 'foo'
GO
SELECT * FROM dbo.TestConversion --this works
SELECT * FROM dbo.TestConversion WHERE Value = 1 --Error converting data type varchar to bigint.
SELECT * FROM dbo.TestConversion WHERE Value = '1' --Same thing, this doesn't work either.
I would expect both the scenarios to work since the view already filters out the bad data.  What's even stranger is that the second query does not work either.  I can only grab an estimated execution plan on both of them (since I can't actually run the query without error), and they are identical.  Based on the estimated plan, the bad data would be filtered first before the WHERE Value = 1 is applied.
Edit: To test the query plans, I changed the CONVERT to TRY_CONVERT instead. The result of the plans are still identical, and it looks like the filter occurs before the conversion:
