Can you re-pierce your ear immediately if you remove the new piercing right away?

Yes but only in the hands of a professional body piercer. You shouldn't do this unless you are using a professional piercing gun, and if you haven't then you should have expected the piercing to be botched in the first place!

If you intend to do it yourself, be aware that unless you use completely sterile equipment, infection is extremely common. Bacteria often enters the piercing (it is a wound after all) and this causes anything from redness and inflamation through to pus seeping from an abcess followed later by scarring of the tissue.

Oh and it can kill you if you get the wrong kind of infection, or have a compromised immune system (i.e. if you had suffered the flu or similar virus beforehand).

For this reason, Answers.com recommends you do not try to do it yourself!