I don't know if it reaches the point where it is unhealthy. But if you are pushing yourself hard every bit of cooling helps (overheating may be your limiting fa

I don't know if it reaches the point where it is unhealthy. But if you are pushing yourself hard every bit of cooling helps (overheating may be your limiting factor, maybe not) and given good air flow your armpits are probably the biggest natural air conditioners on your body. And the cooler you are the less you sweat (and therefore the less water you lose) so during sports or workouts there's no reason to hinder it, even by 1%. The coolest and most comfortable possible workout for me involves wearing a sleeveless shirt, lots of air circulation, and no anti-perspirant. As a bonus my armpits don't stink afterward either. For much the same reason that people don't reek in a sauna. Fresh sweat pouring out of your body doesn't reek. Remember, it's only stale sweat that reeks -- it's the waste products of bacteria that smell, not the sweat itself. (I most definitely do smell body odor on myself+clothing once the sweat is stale, so I'm not noseblind to myself)