Hi margish,
if possible, please contact Apple support about this.
There is an internal issue (I was told without activity since Dec 2016) which hasn't gained much traction yet, as Apple personnel has not reported many devices as affected yet. Unfortunately, I didn't think of asking for the ID. Maybe you could and post it here for reference. The issue relates to reminder notification reliability with iOS 10.
We really need more tracation on this. I have no clue how this issue can stay under the radar for so long while being this serious. I myself was already able to reproduce it on all iPhones I was able to test as of now, which were 1x iPhone 6s, 1x iPhone 6. These were unrelated phones, both running iOS 10.2.
The most convenient method for testing for me has been to schedule a few reminders in a row (with no delay in between) with Siri and let the phone rest. Reminders do seem to work better when the phone resp. display is on.
"Remind me to Test One in 3 minutes."
"Remind me to Test Two in 10 minutes."
"Remind me to Test Three in 25 minutes."
"Remind me to Test Four in 45 minutes."
"Remind me to Test Five in 75 minutes."
In most cases at least one of them doesn't fire, and for me it is often already the second one.
If you do not clear the notifications, you will have a nice overview of which reminders fired and which have not in the notifications bar.
BTW: I have tested this with reminders on a caldav list, on a local list and on an iCloud list. All of which were affected, while, with the few tests I did with iCloud, I had a slightly better hit rate (better reliability) with the iCloud list. (possibly a coincidence)