>I put a 20 seconds pause after the Update. Here is what it gives:
>
>-------------------
>Adding MyTable.FaxPhone-Temporary...
>Updating MyTable.FaxPhone-Temporary...
>
>(1838535 row(s) affected)
>Waiting for 20 seconds...
>Droping column MyTable.FaxPhone...
>ALTER TABLE DROP COLUMN FaxPhone failed because one or more objects access this column.
>-------------------
>
>I have to say I am beginning to run out of solutions. I tried a lot of things and something is causing this behavior which I cannot explain.

Peter suggests to check the indexes (if any index includes this field).

You can check indexes using
sp_HelpIndex SP.

In addition, the field can be used in Check constraint.