It's almost impossible to answer your question without getting into the details of the specific equipment involved. There's also likely to be a difference between what 'works' and what's 'legal.'
First off, there's the size of the feeder wire. NEC has you size that to the horsepower and the voltage. At higher voltages, you might very well already have wire large enough for the higher horsepower.
The starters are another issue. Your overloads will almost certainly be the wrong size, and you'll have nuisance tripping at every start-up. It is possible that the starter will have the range to cover the larger motor- it all depends on exactly what size starter was used.
Now, we've discussed this before, and code rules fail to recognize that there's a lot of fiction in motor nameplate markings - especially when it comes to the horsepower rating. Motor manufacturers routinely re-tag motors to suit the customers' convenience. Since you really have no way of telling what's inside the motor, you need to look at the FLA and the service factor to get some idea as to whether you really have a larger motor, or just how much larger it is.
It would be too easy to blame the manufacturers, and assume they're just a bunch of crooks. That would be unfair. Especially in this day of "energy codes," there are a huge variety of ways to rate a motor's performance. Just for giggles, one might look at the Baldor or Reliance sites. Spec a motor at random .... say, a 10hp, 3600rpm, 230/460v, TEFC motor. Make yourself a little table for horsepower, FLA, SF, and power factor. List the specs for motors of standard, NEMA efficiency, high efficency, and premium efficiency. Maybe also list the specs for similar motors intended for different uses. I think you'll be amazed at just how much those numbers vary.
So, what's the industrial guy to do? Well, that's where we separate the competent managers from the pretenders.
In an ideal world, you would perform periodic Megger tests and vibration tests. You would know ahead of time just when that motor was going to die, and nave the right one on the receiving dock just in time for a scheduled shut-down. The replacement motor would be exactly the same as the old one, and still be readily available.
At the other extreme, you have the place that has been coaxing the same worn-out, abused, obsolete piece of kludge for decades. They've been bypassing controls and jumpering sensors for years, manually pushing the thing until, one day, they let the smoke out. Now, suddenly, they want you to work miracles. You jerry-rig a temporary fix in place, only to find they have no intention of ever letting you fix it right; they're going to run the fix into the ground too.
You can't fix stupid.