I remoted into the app server (not web server, I keep saying that wrong) with the login in question. I logged into SSMS using windows authentication and opened up the database server. I browsed through object explorer on the databases and tables in question no issue. I opened up a few tables with no problem.

The SQL error log's ever so helpful "error" from around the same time of the error (this happened about a minute earlier than the package failure, though) is "Error: 18456, Severity: 14, State: 11".

The SQL Agent log has nothing from around that time.

I need a wall for which to pound my head against.