When I hook the module up to an oscilloscope, I see dips in the power rail (which correspond to the current spikes described on p.18 of this design note) that go low enough (from ~3.27V to ~3V) to brown the module out. That said, I was able to send SMSes was probably upwards of 80% when I ran my tests. What this means is that, in practice, the SIM900 does not always shut itself off as described in the datasheet. It also means that we're not doing a good enough job of providing the module with stable power, regardless of the source of the power.
My guess is that individual modules' results (read: tendency to turn the entire Tessel system off, ability to not brown out just the module, likelihood of successfully sending an SMS, etc.) vary in large part based on variations in the big capacitor near the SIM900 (electrolytics are notoriously inconsistent across...everything). Adding a 10uF 0805 ceramic capacitor to my module at the SIM900's power pin seemed to get rid of the dips. The cap is the brown rectangle in the center of the image below:
All that said, I think I saw the dips more frequently before the module joined the network and when it was transmitting (as opposed to after joining but idle). I'll run more tests tomorrow to see if this is just a figment of my imagination here.