Were Abu Bakr and Alī ibn Abī Ṭālib alive at the same time?
Yes. Abu Bakr and Alī ibn Abī Ṭālib were alive at the same time for 35 years, from AD 599 to AD 634.
Alī ibn Abī Ṭālib was 35 when Abu Bakr died.
Alī ibn Abī Ṭālib
4th Rashidun Caliph (r. 656–661) and first Shia Imam
Born 17 March AD 599 · died AD 661, aged 61 or 62