We have a DTS package utilizing an ODBC connection to and Oracle database to run a few updates. I can TNSPING the Oracle instance but get the error Ora-01034 ORACLE not available when trying to run the package.
DTS packages can run as scheduled tasks, on the server itself, or from a workstation. They may or may not have the same NT Credentials. My guess would be that you're running the task from two different machines (a workstation and a server), where one has Oracle connectivity and the other does not.