This behavior is based on lithium ion battery chemistry, as you suspect. The cutoff is in the controller, not the battery, as Chris Hammond described.
I keep an eye on the voltage indicator -- not the little bars, that's a very rough estimate -- and try not to go below 46 volts. I occasionally get as low as 44V, which is roughly 30% of battery capacity, but performance has dropped considerably by then and drops even faster from there on. Twice, I've gotten down to 42V -- 20% -- and I'm doing almost all of the work. With my 17.4 Ah battery, I can ride 50-60 miles (depends on terrain, wind, etc.) before getting below 44V. Plenty for me, until I get in better shape.
If you need to know how to set the voltage cutoff and a bunch of other useful stuff, check out
this video with
@Tora Harris.