Can batteries charge & discharge at the same time?

A Battery can not be charged or discharged at the same time. However it may so happen that a charger is connected to a battery while the battery is connected to a Load. Let the Charging current be Ic . The Current into or out of the battery be Ib and the Current through the Load is IL.

Batteries are designed to charge and discharge at the same time - this is called pass-through charging.

There are three possible situations:

1. P_in > P_out: there is netto power going into the battery: charging

2. P_in < P_out: there is netto power going out of the battery: discharging

3. P_in = P_out: battery keeps unchanged

   
   

This means you can have a solar panel or some other power source recharging the Voltaic battery while another device (cell phone, tablet, laptop, etc) is taking power from the Voltaic battery. 

If your device uses more power than is being generated by the solar panels, then the battery will eventually drain, but slower than it would normally.