RE: 125 amp alternator
You need a few things to get full output.
Obviously the alternator has to be 100% - all good diodes, brushes, etc.
It has to turn fast enough. 5000 RPM is a good target. You can figure that out by your cruise engine RPM multiplied by the ratio between the crank and alternator pulleys. A 3 or 4 to one ratio is going to be necessary for the tractor engine derivatives (Lehman, Deere, Perks, etc). But you have to be certian not to overspeed the alternator at engine redline.
The belt has to make it turn fast enough. Sometimes they'll make dust if they're slipping, sometimes they'll just glaze over and look just fine. If you have an external fan on the alternator and you can grab the fan and manage to get the alternator to turn, it's going to slip. Above 100 amps, dual belts are a good call if you're using V belts (I don't know what the power transmission capabilities of the flat serpentine belts are).
Using a handheld optical tach isn't a bad idea - measure the alternator speed at full load after it's run for a few minutes.
Then there's the regulator, the field voltage, and the wire between the alternator and the battery. To get full output, the regulator ideally should sense voltage at the battery, not at the alternator (like most internal regulators will). The field voltage is the voltage between the field termal and the alternator case - so if there is any resistance on the alternator ground path, that reduces the field voltage. For example, .01 ohm of resistance (which is very, very little) causes a half volt drop at 50 amps ... so the alternator case is actually at .5 volts, and the field voltage is reduced accordingly. With the alternator running at full output (or as full as you can get it), measure the voltage between the alternator case and the battery ground. Any drop there will result in a drop of output current. A solid ground wire between the alternator case and the main ground buss isn't a bad idea even if there's a solid ground to the engine block.
Finally, there's the wire between the alternator output and the main bus. A remote sensing regulator can compensate for voltage drop here, but only to a point. If you have drop, you're burning up a portion of your power output from the alternator and just turning it into heat. At 100 amps or more, #4 wire with properly installed terminals at each end carrying the alternator output is not overkill.
Temperature of the alternator is important, as FF points out. The magnetic characteristics of the rotor (which is just a spinning electomagnet) get worse as it gets hotter, and it gets hotter the higher the output. Plus the life of the diode assembly is inversely proportional to the temperature. If it's possible to duct some cool engine space intake air (without salt spray) to the alternator, it'll be just that much happier.
Finally (I've rambled too long, I know) -- conversion to a higher amp alternator is more than increasing the diode capacity. The rotor and stator magnetics have to be bigger in order to create enough magnetic flux to induce the greator current. The stator windings have to be big enough to carry the current without melting. Only then does the capacity of the diodes come into play.