No felix, you don't need to restrict yourself like that. You can play any note at any time regardless of the chord being played. What you have to bear in mind is that some of these notes will clash badly with the underlying chord and should be used with care.
Using your example of C major
Playing C, the root, will sound very good. Playing E and G (the 3rd and 5th) will sound great. Playing A, B, D, F (the other major scale notes) will be safe (but perhaps a bit bland). The other available notes must be used with caution, linger on them briefly and then 'resolve' the effect they create by playing a chord tone. It is this dissonance and resolution that creates tension and excitement in music.
A good rule of thumb is to start and end each phrase you play with a chord tone, anything in between is up to you.
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