No, Chase does not let you churn (under most usual circumstances). They used to, it was really sad when that went away.

They currently seem to be limiting most people to opening 1 card every 6 months. You can still get the bonus miles for each type of card, e.g. United Visa Signature, Visa Signature Gold Class, Visa Signature Platinum Class, Visa Business... but they'll only give most people 1 card per 6 months.

Bank of America may say one bonus for eternity, but in my experience this hasn't been enforced.