I strongly agree with the skepticism about breathing having any serious effect on attention.
I have a caved-in chest (about medium), have quite poor fitness and can't swim. My lung capacity is way below average, but I have no problems at all concentrating. My attention span is significantly above average, I don't find reading or writing long sentences difficult at all. I can easily hold large chunks of information in my head if I must. My meditation practice is doing just fine. Concentrating for an hour or more is not tricky at all. (I blame video games. Playing >8 hours/day of high-attention games during high-school does this to you.) Only distractions are tricky, but then only if I'm trying to avoid unease.
Last summer, I did about 2 months of jogging to improve my fitness. I went from not being able to run for 10 seconds to running over 20 minutes without pause. This did not affect my mental capacities in any way I could notice. (Memory might have improved, though.) At best, thinking while running improved, but I never found it hard to listen to audiobooks, even when completely out of breath or exhausted.
On January 4, PJ Eby sent around an email linking an... interesting... website. The claim on the particular webpage he linked was as follows:
This site is part of a sales pitch, so many of the claims are stated in hyperbolic language. I've already noted one factual error: the webpage claims that being underwater triggers the diving reflex, while in fact (or at least, according to Wikipedia) the diving reflex is triggered when one's face is immersed in water colder that 21 °C.
But there is a testable claim here: learn to hold your breath for longer periods of time -- particularly in conditions that elicit the diving reflex -- and you will see increased intelligence. I know that some readers of LW regularly train and test their intelligence, so I offer this as an easily implemented potential method. The possible gains seem to me to outweigh the costs of the training and the low prior probability of the claim.