My electrical professor told our class (on the subject of AC reactance) that if you increase frequency, voltage does not change. He summarized by saying "if you increase the frequency you are just making the cycles shorter and faster but the voltage is still the same."

But, weeks before this class we were told, in order to increase the voltage of an AC generator you can increase the rotational speed... which would mean increasing the frequency (f=N/60).

Is my teacher incorrect in one of these cases or am I taking these two examples out of context?

I suspect you are taking the first statement out of context. He said "the voltage will not change" ... voltage of what? You mention reactance in teh subject - are you referring to, say, capacitive reactance? Capacitive reactance is inversely proportional to frequency, so it definitely changes with frequency. However, at very high frequency the reactance goes to zero, so perhaps he meant above some reasonable value of frequency the voltage across a capacitor is effectively zero and so doesbn't change (practically speaking) with further increases in frequency.