I tried restarting the iPhone, I closed the app in the multi-task bar, I even re-installed the thing, and still it would cut out. It would always take me a while because sometimes they stick some silence in weird paces on the show and it was hard to tell if maybe it was an artistically long pause or something.
Finally, I looked up whether there was anyone else with the same problem, and found this nice article that outlined everything I'd done except for one thing:
Hold the top power button down, then when the power slider appears, press and hold the home button.
This does more of a full reset, clearing caches, brushing out cobwebs, telling the kids to go home. And it worked! Today I wanted to keep running just to listen to more awesomeness, but my 1.5-year-old was ready for home.