Imagine a user sending me some eosio.token to my contract and when I receive the notification, I start a new deferred transaction that fails later.
Can the deferred transaction make the initial transfer fail?
No, the success or failure of the deferred transaction is independent of the action in which it is called.
The only exception is if the deferred transaction would objectively fail in the original action. For example if your deferred transaction would transfer more money from the account than is actually stored there at the time of calling the deferred transaction, then the original action will fail I believe.