Nest customer service is probably your best bet, but I'm going to give you a guess.
In a two wire install the Nest charges itself when the boiler is off, using the difference in voltage between the two sides of the on/off relay inside the Nest itself. A small amount of current is allowed to flow through the charging circuit, and back to the power supply via the boiler's on/off circuitry. This current is small enough to charge the Nest but not signal the boiler to turn on. When the Nest wants the boiler on it closes an internal relay and shorts the two wires together. Enough current flows to signal the boiler to turn on, but there is no longer a difference in voltage to charge the Nest.
It has been very cold in the US recently. My guess is the Nest has to turn the boiler on for long periods of time, probably long enough that its small internal battery has not had enough time to recharge while the boiler is off.
If you are handy you might try tracing the thermostat wiring and look if it would be possible to install a new cable with more conductors so the Nest can have always on power available to it. If you can run the wire (available at most home improvement and hardware stores) a heating technician could switch over to the new cable in a single short visit.