What really matters when you try to push current through a wire is the current, not the voltage. The wire doesn't care what voltage it happens to be sitting at. Only the insulation might care, as too high a voltage across the insulation will result in a breakdown and then an arc.
The current carrying capacity of a wire is determined by the resistance of the wire and how much heat is generated as a result. The resistance of the wire determines the magnitude of the voltage drop across it. Let's say you have 100 feet if wire with a resistance of 0.01 ohms per foot. The total resistance is 1 ohm. If I put 1 amp through that, it will generate a 1 volt differential between the ends of the wire and the wire will dissipate 1 amp * 1 volt = 1 watt of heat. A fuse is just a wire sized in such a way that it will generate enough heat to melt when a specific amount of current passes through it. With proper cooling, a wire can carry an essentially unlimited amount of current. Wire current ratings are generally designed so that the wire can only produce a certain temperature rise at the rated current, under specific environmental conditions. Naturally this will be affected by ambient temperature, insulation (both thermal and electrical), airflow, etc. Bigger wires have a lower resistance and as a result can carry more current with the same power dissipation.
This voltage drop is why long distance power transmission uses very high voltages. Since the power lost in the wires is proportional only to the current, the voltage is stepped up in transformers as much as is feasible to reduce line losses. The end result is that the wires carry a tiny fraction of the current that they would have to if the voltage was not stepped up. Our electric grid is AC for this reason: it's very easy to build a transformer and step the voltage up or down. It's not so easy for DC.