I would expect the limiting factor to be the speed rating for the grease lubricated rotor bearings – easily found from bearing manufacturer's data if you can find the bearing numbers. If the bearing numbers are not to hand, go by the nearest bore size up standard deep groove ball bearing to the output shaft diameter (the shafts are usually a bit smaller than the bearing bore).
If the fan were considered an issue (though I doubt that the lightweight plastic fans used on motors would burst), just take it off & replace it with an external blower – something motor manufacturers offer for motors used with variable speed drives as an option. This would be beneficial anyway, as cooling airflow with the standard "stirrers" is much reduced as the motor speed is reduced with a VFD, but heat generation is not.
A modern motor will be a better option for VFD use than an older unit, as modern "high efficiency" motors run with much closer rotor / stator clearances, are better balanced & will have been designed to take inverter operation into account.
Highest speed I experienced syncronous motors operated at was 12000 rpm – "motor spindle" units on an aluminium router hogging wing ribs out of billets. The motor "shaft" was a 40 ISO taper spindle with a drawbar. The bearings were the limiting factor, life being around 9 months in operation. Precison grade bearings lubricated with Klueber high speed grease that required around 2 days to "run in" – but a bit more extreme application than running a Student !
These were unusual motors – most modern CNC machine tool spindle motors have permanent magnet rotors with encoder rotor position feedback, driven by specialist drives – not your average "run-of-the-mill" squirrel cage motor + VFD.
Nigel B.