I have an old laptop power supply which provides 3.5A at 20V. I want to re-purpose this for another project for which I require 3A at 5V. If I cut the output cable I'm assuming I'll find a +20V wire and a 0V wire? If I chuck some resistance on the +20V to drop it down to +5V will this have the desired effect? Will I end up altering the available current or otherwise messing things up?
You might mess up the current depending on what you want to draw, but you will drop 15V at 3A through the resistors = 45 Watts. You will need power resistors mounted on large heatsinks with a reasonable airflow over them. In short, it's a bad idea.
It would be better to use a power supply better matched to your needs.
But, if I you really want to use this power supply, don't use a resistor. The reason is that the voltage drop through the resistor depends on how much current your device under test is drawing.
It's better to use something like a buck-down convertor or a linear regulator. The linear regulator will have to be well-heat-sunk though, because it will dissipate the 3A *(20V-5V) = 45 W. The buck-down convertors are more efficient but a bit more expensive. You can find them on ebay/amazon/farnell