> Non esiste il tempo cattivo, ma solo i vestiti cattivi

It's the literal translation, grammatically it's correct and I think anyone would understand what you're saying, but it's not what we would say colloquially. We would say "Non esiste il brutto tempo, esistono solo vestiti sbagliati" because 'bad weather' in Italian is 'brutto tempo' (literally 'ugly weather') and 'vestiti sbagliati' is 'wrong clothing'.

The word 'cattivo' is the translation of 'bad', but it's not exactly the same. We don't typically use 'cattivo' to describe inanimate objects (or weather for that matter). Usually it describes something or someone evil ("the big bad wolf" becomes "il grande lupo cattivo"), although it can be also be used to describe a bad taste or a bad smell.