The mentality of "It won't happen to me" has been reinforced since 2008 because we've seen how governments will go to extraordinary lengths to bail out banks. For example, no UK deposit holder has ever lost a penny despite Northern Rock, RBS, HBOS and IceSave all going bust.
Yes, it's prudent for deposit holders to spread funds over several bank accounts. Read the small print though! If you're in the UK, the guarantee is per saver, per authorisation[1] e.g. two accounts in two different banks which are part of the same authorised banking group will only provide single coverage.
However, even this might be an exercise in futility if the deposit insurance fund is empty. 'deposit guarantee schemes may be required to contribute to the recapitalization of the firm'[2]