When did Vasubandhu, the teacher of Dignāga, live? Vasubandhu is known to be the teacher of Dignāga. When did he live? What are the sources that say so?
The problem is that in the oldest known source (Paramartha's "Life of Vasubandhu", compiled around 550 CE) conflicting dates are mentioned. This has lead to several different theories about the exact dates that mark Vasubandhu's life.

According to Wikipedia Vasubandhu lived in the 4th century CE. For this information they refer to this essay where it says:


  it is known that he was a contemporary of Emperor Candragupta I. Ramashankae Tripathi, History of Ancient India (Delhi 1942), gives the dates 320 to circa 335 AD for the reign of Candragupta I, and c. 335-375 for that of his militant successor Samudragupta. This situates the life of our Vasubandhu between 290 and 370 AD. 


However in 1951 Erich Frauwallner, a professor in Buddhism, wrote an article called "On the date of the Buddhist master of the law Vasubandhu" in which he posted a theory that there were 2 Vasubandhus;


the first Vasubandhu was the co-founder of the Yogācāra school and lived  from 320-380 CE, 
and a second Vasubandhu, the author of the Abhidharma-kośa, who lived from 400-480 CE. 


This view has been criticized and doesn't seem to be widely accepted. More info in these papers:


Dan Lusthaus - What is and isn't Yogacara (against the two Vasubandhu-theory)


  Since the progression and development of his thought, however, is so strikingly evident in these works, and the similarity of vocabulary and style of argument so apparent across the texts, the theory of Two Vasubandhus has little merit.

Marek Mejor's - The problem of the two Vasubandhus reconsidered (in favor of the two Vasubandhu-theory)