Actually, no. What you're forgetting is that the planet is *huge*, even the massive consumption of the entire human race is only a tiny fraction of it's total capacity biomass.
We are currently consuming biological resources at somewhere between 1.5x and 2x the rate at which the planet can produce them. Remember it's a living, *growing* system - so long as we consume things no faster than it grows, the system is sustainable. As it is though we're "spending the capital" - not only are we consuming the new growth, but we're also reducing the "base" biomass so that, on average, next year things will grow more slowly than they did this year, and that trend will continue until we reduce consumption sufficiently.
Fishing is probably the most obvious example of this - go find some photos of the docks with fishermen showing their prizes from the 1800s: huge 4-8 foot fish all over the place, while today you rarely see anything over a couple feet. We've been "strip-mining" the ocean for centuries, and as a result the total yields have been falling for almost as long.