Wa`alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
The cut-off point for praying Tahajjud is when the exact the time for Fajr begins. However, its preferred time ends 10 or 15 minutes earlier.
In his response to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
Tahajjud Prayer is part of the Sunnah of the Prophet (peace and blessings of Allah be upon him) which Muslims should follow all the time; it’s not something confined to Ramadan.
It is best to delay this Prayer to the last third of the night. Abu Hurairah quoted the Messenger of Allah as saying, “Our Lord descends to the lowest heaven during the last third of the night, inquiring: ‘Who will call on Me so that I may respond to him? Who is asking something of Me so I may give it to him? Who is asking for My forgiveness so I may forgive him?”
The cut-off point for praying Tahajjud is when the exact the time for Fajr begins. However, its preferred time ends 10 or 15 minutes earlier. We read in the authentic traditions that the Prophet (peace and blessings be upon him) used to pray Tahajjud and Witr, then he would take a brief nap in which he would dose off a little before Fajr.
Having said this, I must rush to add that there is nothing wrong for anyone who awakened to catch up on a few rak`ahs as long as he does so before the time for Fajr starts.
Let us, for instance, say this: These days Fajr starts around 6 a.m., so if a person wakes up at around 5:45 a.m. he can still pray a few rak`ahs if he chooses to do so and they will still be reckoned as Tahajjud.
It should be comforting for us to know, as a principle, that the Prophet (peace and blessings be upon him) said, “This religion of ours is simple and easy to follow; whoever makes it hard will only be overwhelming himself with it, thus being forced to give it up altogether!” (Al-Bukhari).
Almighty Allah knows best.
Excerpted, with slight modifications, from: www.islam.ca.