remember there is no "at the end of the cable" per se, it loops back to the source through ground. Voltage at the source (actually the difference in voltage between the source and ground) tends to push current.
If you connected a million cables from terminal to terminal on the battery, something like infinate amps would flow, the cables, offer little to no resistance to current flow (due to such a large cross sectional area).
So current in a circuit is determined not only by the voltage but by this resistance as well. Thus, ohms law (variation) I = E/R.
So from that you can see that the load, or what your hooking up at the end, determines current flow, no matter how many cables you hook up to it.
Voltage pushes, loads allow current to pass (and determine how much given a particular voltage).
Cables just need to be chosen so that the current the load wants wont be to much for them.
EDIT: But technically, to fully answer the actual question, + pushes, - (ground) pulls, (equally) and loads resist. The device will allow just what it needs if it's working right, no matter how big or how many cables you have going to it. Too much is ok (cables that is, not voltage).
I bet I'll hear about electron-flow now. But I was just trying to not be too confusing, so I stayed with the hole-flow convention.