Were Muhammad and Alī ibn Abī Ṭālib alive at the same time?
Yes. Muhammad and Alī ibn Abī Ṭālib were alive at the same time for 33 years, from AD 599 to AD 632.
Muhammad was 27 when Alī ibn Abī Ṭālib was born.
Alī ibn Abī Ṭālib was 33 when Muhammad died, aged 61.
Muhammad
Arabian political leader and the founder of Islam
Born 20 April AD 571 · died 8 June AD 632, aged 61
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