Can all hadiths be trusted?

Among the hadiths attributed to the Prophet, some are authentic with no doubt as to their attribution to him, while others are weak and their attribution is questionable. Both categories are known to scholars, and books have been compiled that gather the authentic hadiths, such as Sahih al-Bukhari and Muslim, while other books include both the authentic and other types. Scholars have made great efforts in distinguishing the authentic from the non-authentic, such that the matter is not hidden even from young learners.