It can happen.. How big an amp?  A class A audio amp running
high power draws quite a bit of current.  Your electrical system is
ok most likely. The alternator seems to be working.  You just ran the
battery down. The battery could be getting a bit weak.. BTW, usually
a lighter plug is not a very good place to connect high current gear.
The wire is thin, and the voltage drop fairly high.
I run ham radios in my cars, trucks, and have to consider current
draw. I usually carry two batteries if I'm going out of town.  My radio
runs about 100w output, and draws appx 20 amps at full power.
I run my power wires direct to the battery using thick wire. Less
voltage drop, which means more usually "on" time before the
voltage drops too low for the radio to work correctly.
Using thick wire to high power audio amps is a pretty good idea.
You'll have better dynamic range as the amp is not seeing as large
a voltage drop on current peaks. You are drawing some hefty
current, or your battery is about half shot with a dead cell, etc..
BTW, every time you run the battery down like that, it loses some
capacity.  You might want to get it tested..
MK