No alarm or silent during certain times
Since I also realised it happens with widgets, I worked out it's due to battery saving. 

P8 has option to allow apps to run even screen off (To find this, go to Settings -> Power saving -> Protected apps), and it appears this also means in background (so they continue to run even if they're closed)! 

It's fixed my email widget and my SMS/bandwidth measuring programs. 

It has fixed my alarm, but not my 'silent time' application, but the phone has this option built in. Despite that, I will test another application to see.