Can we possibly have too large wire to drop voltage over a long distance?
More the current flowing through the wire, more will be the voltage drop in the wire. All practical wires have non-zero finite resistance which means you will get a voltage drop equal to current x wire-resistance.
Now, if you need to power up an appliance of lets say 100 W and you are presented with 2 options:

10 V and 10 amps

100 V and 1 amp


Lets say wire resistance is 0.1 ohms. Voltage drop in first case will be roughly 1 V which is 10 % of the source voltage. In second case, the voltage drop will be 0.1 V which is 0.1 % of the source voltage.
Thus if both options are feasible, its better to go with second option.
Also, (current x current x resistance) heating loss will be lesser in the second option because of lower current.
That's the reason power plants prefer to send the power over high voltage and lower currents. That reduces wastage.