Sometimes a hadith can have a sahih chain, but still be rejected.
In Tafsir Jalalayn verse 7:190 we find a peculiar narration:
Translation: But when He gave them a righteous child they ascribed to Him associates (shurakā’a a variant reading has shirkan meaning sharīkan ‘an associate’) in that which He had given them (by naming it ‘Abd al-Hārith ‘servant of al-Hārith’ when it is not right to be a ‘servant’ ‘abd of any one but ‘God’ sc. ‘Abd Allāh but this namesake ‘Abd is not an association of another with God in terms of servitude for Adam was immune from a sin such as associating others in worship with God. Samura b. Jundub reported that the Prophet (ﷺ) said ‘On one occasion when Eve gave birth — all the children she bore had failed to survive — Satan visited her and said to her “Name it the child ‘Abd al-Hārith and it will live.” She named it so and it lived. This affair was the result of Satan’s inspiration and his doings’ reported by al-Hākim who deemed it the report ‘sound’ sahīh and also reported by al-Tirmidhī who considered it ‘fair-uncommon’ hasan gharīb;) but exalted is God above what (they the people of Mecca) associate (in the way of idols; this sentence is consequent a supplement to the one beginning with khalaqakum ‘He created you’ so that what comes between the two is a parenthetical statement.) (translation source)
This narration has a sound chain, but the matn itself is merely from the Israiliyyat of previous generations. When narrations from the Israiliyyat contradict our Shari’a principles, we reject them. This hadith contradicts the ‘isma (infallibility) of prophets, especially of committing shirk. So this narration is rejected, despite having a sahih chain.
This is also a great example of why studying ulum al-Qur’an is so important as well.