Do chaperone proteins misfold?
Chaperone proteins are still proteins and they can certainly misfold just like any other. If that happens, it will either be assisted by another chaperone and given time to fold successfully or it will be destroyed. If this is happening too often and the amount of chaperones drops too low or the number of unfolded or incorrectly folded proteins becomes excessive*, the unfolded protein response may be triggered and, if it does not resolve the issue and the cell remains stressed, the cell will undergo apoptosis and die.

Some chaperones, especially heat-shock proteins, may be more resistant to misfolding. This is true because they need to be able to withstand noxious conditions that denature other proteins. Not all chaperones are resistant to heat though and many are no more intrinsically resistant to denaturing.

*  It's not uncommon for a "bad" batch of proteins to be created. This can happen with a transcription error during mRNA synthesis since each mRNA molecule is read by many ribosomes. This can naturally happen with chaperones as well.