How does resonance fail in approximating chemical structures?
Resonance structures are one model that explains bonding, but not a very good one

Resonance structures are widely misunderstood. Individual resonance structures are not real compounds that exist, but an approximation that assumes all structures are made from bonds consisting of two electrons. The true structure of a compound is thought to be an average mixture of the various possible resonance structures consisting of simple bonds. 

The archetypal example is benzene. If we model benzene as a structure consisting of "normal" 2-electron bonds, we get a ring consisting of alternating single and double bonds. This contradicts the observed structure where all the bonds are the same (and are somewhere between the character of single and double carbon-carbon bonds). The resonance model of benzene consists of two structures with the normal double and single bonds swapped, giving an average overall structure where all the bonds are an average of single and double bonds.

For structures with more complexity than benzene, the resonance structures are more complex and frequently don't make equal contributions, nor do they do a good job of predicting the real structures. Molecular orbital based approaches (which recognise the difference between pi-bonds and sigma bonds and do a much better job of recognising the fractional character of many real bonds) are far better.

That is why resonance structures "fail". They make drawing simple structures easier, but they don't do a good job of predicting real structures.