It is known that Imam Muslim did not narrate a single hadith through his teacher, Imam Al-Bukhari, in his Sahih. It was documented by Imam Adh-Dhahabi in Siyar A'lām an-Nubalā' 1/87, and he elaborated that they differed in their conditions of what would make it to their Sahihs. However, it is not known for sure why he elected to do so as he never mentioned it himself. Hence, all that follows is an opinion, not to be taken as Imam Muslim's expressed view.
One of the reasons for not including hadiths through Imam Al-Bukhari may be that Imam Muslim had already written most of his Sahih (239 A.H.-253 A.H. to compile, classify, and write according to the best estimates) by the time Imam Al-Bukhari had arrived in Nishapur (around 250 A.H.). See Siyar A'lām an-Nubalā 13/566 (Arabic only) for more information.
Imam Al-Bukhari was not the only teacher that Imam Muslim did not narrate through, e.g., 'Ali ibn al-Ja'd, 'Ali ibn al-Madīni, Mohammad ibn Yahya Ath-Thuhli, etc. He narrated through 210 teachers, which had a high number of teachers that are common to Imam Al-Bukhari. So another reason may be that Imam Muslim took the hadiths directly from their common teachers. This achieves a higher chain of narration (al-isnād al-'āli, Arabic: الإسناد العالي), which was (and still is for matters like ijāzah) highly sought and cherished. A higher chain of narration means fewer narrators between al-mukharrij (the person documenting the hadith) to the Prophet ﷺ (hadith marfū') or to a companion (hadith mawqūf). Refer to Sharh al-Mandhūma al-Bayqūniyya 3/24 (Arabic only) for more information. It is worth noting that the average chain narration for Imam Al-Bukhari is five, whereas the average for Imam Muslim is six.
Of course, without documented evidence from Imam Muslim himself, all one can do is speculate.