Can you become re-infected with the same std, after already getting it treated?

Depends. Yes, mostly... Gonorrhea, chlamydia, syphilis, etc... Can be acquired any number of times. Their is no limit and no protection from repeat infection. Hiv, HSV and likely hpv are permanent and dont ever go away. Of course, there are more than one type of hpv and HSV (herpes) and you can get all of them possibly.