Many times - no problem. I would often have a monthly rental out in Chicago and pick up weekend rentals in all sorts of cities. Never an issue anywhere.

And just in case someone later searches on this with a slightly different question, I've also had two cars out of the same location at the same time on a couple of occasions. Once from a HLE near my house and once at an airport location where we were traveling to a wedding. On all of the rentals, the car was in my name, but both of them had a CDP code that permitted my spouse to drive the cars.

(I'm not convinced that they specifically checked that the CDP permitted the add'l driver, but from a liability perspective, I wouldn't have done it without knowing I was covered. We would have just rented the 2nd cars under her name.)