If you only have 1 Chase card so far, I would say you still have a decent chance of getting approved for 2 more Chase cards, regardless of the other recently opened credit lines.

Having said that, you'll probably be rejected initially and have to go through their reconsideration line. It's definitely not 100% that you'll be approved via reconsideration, especially if they cite "too many recent inquiries". But Chase apparently has been fairly flexible lately, so I'd say it's worth a try (if you don't mind the additional hard pulls).

Today, I applied for both the business card, Chase Ink Bold, and the personal card, Chase Sapphire Preferred. (I already have a Chase United card.) I applied for the Ink card online and received an email that said they'll let me know within 30 days. About an hour later, I applied for the Sapphire card on the phone and the agent said I'll get a snail-mail letter within 2 weeks.

After I got these responses from Chase, I applied for the Amex Gold and was instantly approved. (I already have a Costco Amex biz card.)

My credit score is 799. The credit limits on my 3 existing cards are in the 10's of thousands. I have no delinquencies, ever. It's been 5 years since I applied for a credit card.

Any idea what's up with Chase? Waiting 10 days in one case, and 30 days in the other seems over-the-top. Who has experience with Chase and can shed some light?

More common expert advice is this:

1) Have a goal for the cards you apply for
2) Wait till your score bounces back to 750 or so
3) Wait 90 days between app-o-ramas because the impact of the hard pulls will lessen
4) Make sure you can meet the spend requirements for your new cards
5) Go for it!