I edited this information out because this was about 3,000 words before I cut it down, but interestingly there are examples of amphibians and reptiles today that use both types of breathing when exerting energy in heavy exercise, such as escaping from predators.

I imagine given how each system has an original, separate purpose, it's kind of sensible for lungfish to do that, and they evolved a little more recently than what we're learning in this article, about 277mya, so I guess it fits in place nicely. It doesn't really fit the description of convergent evolution, though lungs did evolve several times separately =)