I would not say (to paraphrase) 'If bad things happen that proves there is no God'. But, I would say, that 'bad things happening' does disprove certain gods; this is because the definition of certain gods run contrary to 'bad things happening'. The Epicurean paradox, I believe, runs contrary to the Christian god mostly described as being 'all loving, all powerful, etc...'.
To look at a sillyexample. "If beard exists, it proves someone has chosen to grow beard--it doesn't prove a barber does not exist". If my definition of my God was: A god that is all powerful, loves the clean shaven as beards go against his very nature, beards represent highest moral wrong, he directs his holy spirit scissors at will throughout the known universe, loves all infinately, but those with beards will burn in hell fire for eternity. Now the fact that beards exist would potentially mean my god does not exist...eg if he infinately loves some with a beard, he would not want them to burn in hell, my god has the power to cut a beard off so why would he not?
An atheist can only really examine your God hypothesis by the characteristics you assign to the god if no direct evidence for your god exists.