Praying both tarawih and tahajjud in the same night

Answer

There is no prohibition in the hadith of 'A'ishah against doing more than 11 rak'ahs (whether tarawih, tahajjud, qiyam al-layl, or qiyam Ramadan). A prohibition requires a prohibitive principle; it cannot be deduced from a narration that simply mentions what the Prophet did. On the three nights the Prophet prayed tarawih with the people, narrations differ on the number of rak'ahs: some say 20, some 13, some 11. The majority of scholars say there is no rigorously verified (sahih) report on the exact number -- however, the hadith of 'A'ishah is sahih. It is possible she was referring to night prayers other than tarawih; the 8 rak'ahs prayed before the 3 of witr are all called witr in that usage. Tahajjud, technically, is any sunnah prayer performed after 'Isha, after waking from sleep -- so even tarawih prayed then is technically tahajjud. Can one specifically intend tahajjud after tarawih? The majority of the fuqaha' say yes; Ibn Qudamah al-Maqdisi and al-Thawri are cited supporting no harm in it. To summarize: there is no prohibition against praying both; one understanding of the hadith is she meant tahajjud, not tarawih; and the overwhelming majority of jurists agree on the permissibility of tahajjud after tarawih.