majenko:
Now imagine you have two regulators in parallel. They're both going to be making independent decisions about the same voltage. They'll both decide to raise their voltages or lower their voltages, and the net result will be too much or too little voltage all the time, so they'll then try and compensate again. The two regulators will be fighting against each other to gain control of the voltage, and neither will get control. They'll just be getting in a complete mess.
Actually, no!
Presuming the regulators are properly stabilised (and if they are not, then they are useless as regulators in any configuration whatsoever), their set-points will differ; one will be higher than the other. If their set-points were the same, then the resistance in the common wiring would equalise the currents they supplied and this is in fact used in some cases and works perfectly well.
In practice however, the regulator with the higher set-point will attempt to raise the voltage to that set-point, higher than the other which will then turn itself off. The regulator with the higher set-point will be providing as much current as it possibly can, and either overheat or safely current limit, depending on its design. If the latter, it will fail to regulate and the regulator with the lower set-point will simply make up the deficiency in current and be relatively lightly loaded.
Tobster86:
Specs for the Arduino indicate a recommended input voltage range of 7-12V; surely the best I could do is a (presumably rare) 7V regulator, then throwing away 2V with the Arduino's regulator?
As Fungus just noted, an LM317 would - if you wanted to do it the hard way - regulate to 7V quite well
Tobster86:
I understand the Mega 2560 runs on 5V; what's the best configuration of components to step 24V down to this?
A switchmode regulator of course.
Tobster86:
Ideally I don't want to replace the regulators on the Arduino boards themselves due to the number involved in my project.
You do not seem to have a choice. You need the supply to be regulated. How many are you using anyway? This is surely, the number of complete systems you propose to construct, not a number of them you would propose to use in a single design?
Sigh! As always, perhaps you should describe your intended project and take advice on the best way to approach the whole design.