I've had a Chase Freedom since 2007 (back then, I got ~$300 to sign up, plus they gave you $50 bonus every time you'd hit $200 in rewards.) Recently, they were giving out $200 to spend $500, and $25 to add an authorized user. I applied and was rejected (derp!) I then closed my card and applied the next day and was instantly approved.
I've actually opened two Chase cards (Freedom and Ink), two Barclay cards (one is the awesome Sallie Mae), a Citi Double Cash and an Amex Blue Cash Preferred in the past ~4 months. My credit score is down all of 12 points from a high in September/October. That will be temporary though. Six months from now, it will likely be even higher.
$600 cash and 50k miles in bonuses and now I get 5% back in almost every category.
TL;DR Probably not OPEN at the same time but you could close your current one and re-apply, depending on fine print. (Chase restricts bonuses to once every 24 months per specific card.)