Chase doesn't have an official limit on the number of Chase cards one person can have and many TPG staff have had eight or more Chase credit cards at the same time. But Chase does limit the amount of credit it will extend to you. As a general rule of thumb, you can be approved for one personal card and one business card within 90 days, but even that's not a hard-and-fast rule.
One thing to keep in mind is that there are also different application rules for specific Chase cards. You can only hold one Sapphire card at a time. You can only have a single personal Southwest credit card; however, you can hold both a Southwest business and personal card, which is a great way to earn the Southwest Companion Pass.
And if you open the Ink Cash, you're still eligible for the Ink Business Unlimited Credit Card and Ink Business Preferred Credit Card.
For more details on Chase's application rules, read our guide on credit card application restrictions.