Shaykh Dr. 'Ali Muhammad as-Sallaabee writes,
It was narrated that Jaabir رضي الله عنه said, "I heard the Messenger of Allaah صلى الله عليه وسلم say: 'Islaam will continue to prevail during the time of twelve caliphs.' Then he said something that I did not understand, and I aksed my father: 'What did he say?' He said: 'All of them from Quraysh.'" 
According to other versions, he صلى الله عليه وسلم said: "This religion will continue to prevail and be strong during the times of twelve caliphs." 
"The peoples' affairs will continue to be prosperous during the reigns of twelve men." 
"This religion will continue to prevail until you have had twelve caliphs, under all of whom the Ummah will be inited." 
A similar report, via al-Aswad Ibn Sa'eed رحمه الله from Jaabir رضي الله عنه, says: "He added: 'When he returned to his house, the Quraysh came to him and asked: "Then what will happen?" He replied: "Killing."'" 
The Twelvers cling to this text and quote it as evidence against Ahlus-Sunnah. This is not because they believe in what is mentioned in the books of the Sunnis;  they quote the ahaadeeth they believe in as evidence against them.
However, by examining the text with neutrality and objectivity, we find that it describes these twelve men as becoming caliphs. It also states that during their time, Islaam will be strong and prevalent, people will be united behind them, and the Ummah will still be in a good state.
None of these attributes is applicable to those whom the Twelvers claim as their Imaams. None of them became caliph except Ameerul Mu'mineen 'Ali رضي الله عنه and after him, al-Hasan رضي الله عنه for a short time. According to the Shi'ah themselves, the Ummah was not prosperous or dominant during the time of any of these twelve Imaams. On the contrary, the Ummah "continued to deteriorate; it was ruled by evildoers and even by disbelievers,"  and "the Imaams themselves concealed their religious affairs by means of taqiyah."  
The era of Ameerul Mu'mineen 'Ali رضي الله عنه, when he was in a position of power, was 'an era of taqiyah', as clearly stated by their Shaykh al-Mufeed.  He "could not bring out the Qur'aan or rule in accordance with many Islamic rulings," as clearly stated by their Shaykh al-Jaza'iri.  He was 'forced to go along with his companions and appease them at the expense of religion,' as stated by their Shaykh al-Murtadaa. 
Thus, this haeeith has nothing to do with their claims at all. Rather, it is a foretelling from the Prophet صلى الله عليه وسلم that Islaam would continue to prevail during the era of these people. The era of the Rightly-Guided Caliphs and the Umayyads was an era of power and strength. 
Shaykhul Islaam Ibn Taymiyyah رحمه الله wrote: "Islaam and the rule of Islaam were stronger and more far-reaching at the time of the Umayyads than after their time."
He quoted as evidence the hadeeth: "This matter will continue to prevail during the time of twelve caliphs, all of whom are from Quraysh." 
Then he said: "This is how it was. The caliphs were Abu Bakr رضي الله عنه, 'Umar رضي الله عنه, 'Uthmaan رضي الله عنه and 'Ali رضي الله عنه, then those who took over (under whom the people united and who had power and strength, namely Mu'aawiyah رضي الله عنه and his son Yazeed), then 'Abdul-Malik رحمه الله and his four successors, among whom was 'Umar ibn 'Abdul-'Azeez رحمه الله. After that came the decline that has continued until today."
Then he said concerning the hadith "...all of them from Quraysh,": "This means that they are not limited to 'Ali رضي الله عنه and his descendants. If they were limited to 'Ali رضي الله عنه and his descendants, he would have mentioned something to refer to that exclusivity. Don't you see that he did not say: 'all of them from among the descendants of Isma'eel عليه السلام,' or from among the Arabs? If they had been distinguished by the fact that they were all from Banu Haashim or from among the descendants of 'Ali رضي الله عنه, the hadeeth would have referred to that. However, since he said that they would all be from among Quraysh in general, it is known that they were from Quraysh, not from a particular clan. Rather, they are from Banu Taym, Banu 'Adiyy, Banu 'Abd Shams and Banu Haashim. The Rightly-Guided Caliphs رضي الله عنهم were from these clans. 
So there is nothing left of the description to match the description that they want except the number alone, which means nothing."