It's actually almost the opposite situation, in theory. An incandescent lamp provides low voltage drop, unless it's heated up. It takes a fair bit of current to heat one up enough, that it lights. As it heats up, it has a higher resistance to current flow. That's why incandescent lights usually blow when you first turn them on; low resistance equals high current flow, so they blow like a fuse.
LEDs are a different animal. Once they reach a certain threshold current, they try to force a fixed voltage drop across them. For a basic red electronics LED, that voltage is roughly 1.5 volts. They effectively become a variable resistance in the circuit.
"Breakdown voltage" is more of a thing for Zener diodes, which are designed to let current through once they hit a certain voltage. If you hit breakdown on a regular diode, or a LED, then it's usually followed by "burnt chip smell"
Leakage current is generally pretty damned low, though it can obviously be significant in some applications.
I actually went to school for this crap, though I never use it these days
*EDIT* Actually you're right about the 'breakdown voltage' thing. I forgot that they also use the term when talking about both forward AND reverse current.