Participation of Pi-bonds and lone pairs in resonance
You have probably misunderstood what the sentence actually means.

It means that only one, either $\pi$ bond or lone pair will participate in resonance if the atom has both. 



For example, in pyridine, the nitrogen has lone pair as well as is attached with a $\pi$ bond. 

While resonance, its $\pi$ bond participates in resonance because it (and not the lone pair) is in the plane perpendicular to the molecule.



Observe that the lone pair remains at the place it is meaning it doesn't participate. 



Conclusion: The pair of electrons present in the orbital perpendicular to the plane of molecule participates in resonance if it is present.