# Nontrivial obstruction cocycle whose class is trivial

Are there any interesting examples of relative CW complexes $(X,A)$ where a map $f \colon X_n \rightarrow Y$ into an abelian space has the property that the obstruction cocycle $c_f$ is nonzero, but the class $[c_f]$ is zero, so the restriction of the map $f$ to the $(n-1)$ skeleton can be extended to the $(n+1)$ skeleton?