Just thought I would add our data point:

My SO applied for the personal plus card (50,000 points) on January 8, 2013 and was inta-approved (credit score of ~730).

He applied for the personal premier card last night, June 15, and was insta-approved as well (credit score of ~760).

Apparently the fact that he already had a personal plus card didn't raise any red flags. However, these are his only Chase cards.