I, too, have had several instances of my bike not starting. It usually occurs when the bike is cold, or after it's been ridden but has been sitting for a while (warm).
I found that goosing it with a bit of throttle when the start switch was pressed the second time always resulted in a good start.
However, in reading this thread, I was struck by a new member's posts - Ryanizzle, starting at post #73. He suggested that the fuel choke knob should be engaged during the start sequence, in accordance with the User's Manual.
YMMV if you don't have a Bonnie, but it would probably behoove all of us to actually

.
I immediately checked mine and wah-lah, on page 46, it says that the choke should
ALWAYS be used on a cold or partly warm engine, even if the OAT is >77*F/25*C (emphasis mine).
WH_A_A_AT????
Using the choke knob is the equivalent of goosing the start sequence with a bit of throttle.
The manual also says to leave the throttle completely closed during the start.
I haven't seen a definitive cure for these start issues, so in the future I'm going to start the bike the way Triumph recommends and see if that takes care of *my* start problems.
One final note: there are limitations on the starter switch. My manual says don't run the starter for longer than 5", then you have to have a cool down of 15". If you don't do this cool down, the starter motor will overheat and cause a decrease in battery power.
And if the battery power drops, maybe the ECU detects a ripple in the Matrix and aborts the start sequence...?
Again, YMMV. Regardless, it wouldn't hurt to check your owner's manual for the proper start procedure, not what we all *think* it is. I was guilty of that!
My thanks to Ryanizzle for the suggestion, although I think he has left the Forum.