Question Summary:
If I make an intention to fast at night, but I did not wake up for suhūr, can I choose to not fast?

Question Detail:

If I make an intention to make the sunnah fast of Monday or Thursday the night of, but I do not wake up for suhūr, can I choose to not fast, even though I believe I could still fast?

Answer :

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In the Name of Allāh, the Most Gracious, the Most Merciful.
As-salāmu ‘alaykum wa-rahmatullāh wa-barakātuh.

If you have made an intention at night, then your fast will automatically begin with dawn and you do not need to make another intention at that point.  Hence, if you wake up after dawn, then you are already in a state of fasting since your intention from the night before was valid.  Therefore, you may not break your fast at that point or choose to not fast.
However, if you wake up before dawn, then you may change your intention and choose to not fast since your fast has not yet started. 
 
ثم يباح الفطر في صوم التطوع بعذر وهل يباح بغير عذر بشرط القضاء ذكر في المنتقى أنه لا بأس بذلك وذكر أبو بكر الرازي عن أصحابنا رضي الله عنهم أنه لا يباح وتعارضت الآثار في ذلك فالأخذ بالمحرم أولى احتياطا
(الفتاوى الولوالجية، كتاب الصوم، الفصل الثاني: 1/226؛ العلمية)
 
And Allāh knows best.
Muftī Abrar Mirza
Chicago, IL (USA)
Under the Supervision of Muftī Ebrahim Desai (South Africa)