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From the Sunni perspective Sahih al-Bukhari is renowned as to being the most reliable and authentic hadith source available. From the Shiite standpoint the fact that it has a large quantity of hadiths that are very similar, both in meaning and usage of words to Shiite hadiths, has made it a source that is legitimate to refer to.
Nevertheless, this doesn’t mean that if a hadith could not be found in Sahih al-Bukhari, it isn’t authentic at all; for even the Sunnis don’t see it this way either, otherwise they would have to restrict themselves to the ahadith Sahih al-Bukhari offers when it comes to jurisprudence, theology, etc. A short glimpse at the methodology of their scholars, which has been maintained up to now and today, confirms the fact that they have many authentic and valid references other than Sahih al-Bukhari.
At the same time, Sunnis cannot validate every hadith in Sahih al-Bukhari because that will entail them accepting the hadiths that imply that the Quran has undergone alteration! 
Considering the mentioned facts, it must be said that: Although it's true that Bukhari has not mentioned the mutawatir Hadith of Thaqalayn in his book (a mutawatir hadith is one that has been narrated so much and by so many different narrators that it leaves no doubt that it has indeed been said by the infallible), this doesn’t undermine the validity of this hadith, even according to Sunni standards. The reason to this is that first of all, the Hadith al-Thaqalayn can be seen in the other Sihah (the veritable Sunni books of hadith/al-Kuttub al-Sittah) and other reliable Sunni hadith sources, for example:
1. Sahih Tirmidhi, vol. 3, pg. 328, hadith 3874. 
2. Sahih Muslim, vol. 7, pg. 123. 
3. Sunan Darami, vol. 2, pg. 432. 
4. Musnad Ahmad Ibn Hanbal, vol. 3, pg. 14, 17, 26 and 59; vol. 4, pg. 371.  (An important note is that this book is known to be an essential book in the Sunni school of thought because it has been compiled before Sahih al-Bukhari).
Second of all, Al-Mustadrak Alaa al-Sahihain, by Hakem Neishaburi, which completes the Sahih al-Bukhari and Sahih al- Muslim, mentions this hadith, while stating under it: " الشیخین و لم یخرجاه هذا حدیث صحیح الاسناد علی شرط "  . Meaning that this hadith conforms to the conditions of authenticity of Bukhari and Muslim, but these two scholars refused to bring it up! (However as mentioned before, Muslim has included this hadith in his Sahih, but with a different chain of narrators).
Based on the above, we cannot perceive that Bukhari refused to add the hadith because of its authenticity but rather didn’t want to mention the hadith due to political and religious bias.