As-salamu 'alaykum
There is a lilttle ikhtilaf among the 'ulama on this topic. If you listen to the speech by Mufakkir al-Islam Shaykh Pir 'Abd al-Qadir Jilani about the hadith that is used to clear Yazid (and make him jannati) [The alleged Hadith of Constantinople – Part 1], it shows that 'ulama allow some ahadith in the sahihayn to be "not fully sahih". Likewise, I have met other reliable sunni 'ulama (muftis) too, that shares this opinion, that there are some ahadith in the sahihayn that may not be classified as sahih, while others say - and what is the common thought among people - that they are all sahih.
The speech of Mufakkir al-Islam is on the frontpage at YaNabi.com right now.
In my opinion, it is better not to discuss this topic too much in front of common people, as it is too complicated and requires a higher level of understanding. If we open this discussion it opens the doors for salafis/shi'as to start there fitna, and reject things in sahih ahadith here and there (... shi'a already do). There may be more important topics to cover...
Allah Hafiz
Ibn Amin