Not sure, but I wonder if you guys are talking about two different things.
gabbott, I believe, is referring to deliberately hitting the "snooze button" - that is, swiping left to silence the alarm and start the snooze timer.
The original poster, I believe, is referring to sleeping through an alarm. I suppose (but haven't tested) that at some point the alarm app will silence the alarm on its own (rather than blaring all day as the "legacy" alarm clock on my nightstand does). Once that happens, I don't have any idea what comes next - does the alarm app automatically go into snooze mode and then sound the alarm again at the appropriate time? If so, does it automatically repeat this process?
I don't know.
In Clock Settings -> Alarms, there is a "Silence after" value which appears to default to 10 minutes. My guess is that that is all you get. The alarm sounds for 10 minutes and, if you don't deliberately swipe left to start the snooze timer, you are then out of luck.
OP - you might want to change the "Silence after..." value to something a bit longer (or, simply "Never") and/or change your alarm ringtone to something more effective; there are many to choose from or you can download one of your own. Also, go to Settings -> Sound and make sure you have the Alarm volume cranked up.
When I was in my teens/20's, I would sleep through anything - our neighbors used to complain about my alarm going off while I didn't hear it at all even though it was inches from my head.
These days, I never hear my 5:00 AM alarm, because I nearly always awaken on my own 5 - 15 minutes before the alarm is due to sound.
