President James Garfield would probably have survived being shot if not for the terrible medical treatment he received. His doctors repeatedly stuck their unwashed fingers into his wound, accidentally punctured his gallbladder, and denied him adequate nutrition. Even still, Garfield had such a strong constitution that it took more than two months for him to die after being shot.
Thus, whenever doctors deemed that a person was sick due to their blood creating an imbalance, they would drain "a bit of it" by means of leeches or a puncture, a procedure that came to be known as "bloodletting". In Washington's case, he asked for it, much like today's people who pop a pill without going to the doctor (sometimes it works, sometimes a doctor visit is better).
Thankfully, during the 19th century, people started to question its validity, and the medical community stopped gradually its usage. Nowadays, it's reserved to very, very specific cases, but with so many precautions never even imagined long before.