IME, you're not being declined due to open accounts, but too many recently opened accounts. Chase is infamous for this - 5 new accounts in 24 months and they'll decline you no matter what your credit rating is. Others may be following suit. I've gotten picky about opening new accounts. They have to be worth it.
I will often keep cards or downgrade them to a free card, unless they're an account you can reopen in a couple years to earn the bonus again. The Citibank American Airlines turned into a Citi Double Cash. The Chase Sapphire was downgraded to a Freedom. It helps with your average card age (higher is better) and, assuming you don't load them up, it lowers your utilization rate.