If you demand too much current, the voltage drops to a level you can't use.
And/or the cell gets hot enough to destroy itself! Short out a high-capacity cell with low internal impedance, and it's like a stick of dynamite going off.
NiCads are my favorite choice for these applications. They're cheap, easily available, durable, not picky about charging, relatively compact, relatively high capacity, and can stand wickedly high currents (some can take 100A in short bursts, your 4A draw won't be a problem at all.)