The charging chip (BQ24297) will not connect the battery if the battery voltage is <2.5V (battery short circuit protection).
Below 3.0V, the chip will only trickle charge the battery at 128mA so as not to overheat the potentially damaged battery (precharge).
Between 3V and 3.5V we are supposed to get all the current you can from the charger (up to 2A by default settings).
However, the charging chip will let the system draw current from the battery if the battery voltage is above 3.0V when the charger is not powerful enough to supply the CPU and system demand. So, if the CPUs are running and pulling more power than your power supply can provide, the battery could still discharge down to 3.0V. The operating system should do a shutdown when the voltage drops below 3.5V though.
Unfortunately, the pyra boots up when a power cable is plugged in. Is there a situation where we can't charge a battery when the charger and the battery are not strong enough to boot the operating system? Once the system boots, you can then shutdown, or leave to charge whist idle. I guess this happens with the battery < 3.6V.