What did the voltage drop to?
If you look at the soshine diode photo I linked to, they use two diodes, which switch over as the voltage drops to keep the battery operating at near 3.0V, I can only imagine the same thing being used in the LiFePO4 batteries from Tenergy, as the battery loses its charge you can't keep using the same diode or you will reduce the voltage below 3.0V
The two diodes shown in the picture forward different amounts of current. The Schottky rectifier forwards 1A and the passivated rectifier forwards 1.5A, the battery is listed at 1A max discharge. Perhaps this is the case with the Tenergy, as the voltage drops, it switches to a different diode, which forwards less Amperage. Could explain why the battery stops working after 20 minutes or less, the voltage would drop very fast drawing high current loads, and continuous high discharge rates and deep discharges will reduce the battery capacity greatly over time.
That's my guess, unless someone wants to rip one apart and show us the voltage regulators (I don't own any)
some info from other forums:
Tenergy LiFePo4 750 mAh 6V battery pack testing - RC Groups
All Battery.com LiFePo4 RCR123A - CandlePowerForums
LiFePO4 123 Cell Usage Experience? - CandlePowerForums