Would it be possible to have a text box fade in, fade out, and then have another (different) text box fade in - all in the same slide?

Almost like a slide within a slide?

Yes, you have to arrange the sequencing of the animations. Select the initial text box, apply the custom animation you want. Then select the second box and apply the animation. For the second text as a start option you can choose "after previous" in order to create a flow.