I have this problem with my iPhone 6 Plus too. It also plagued me on and off with my 4S.
Sometimes I will receive alert tones, other times I won't. I'm sitting here tonight texting back and forth with my girlfriend. The phone is right next to me, the volume is on full and I don't have "do not disturb" switched on. Everything is set correctly in the notification settings. When I get a text, sometimes it makes a sound, other times it doesn't. I have to keep checking my phone and sometimes there will be a new text that neither made a sound or displayed on the lock screen. The same thing happens on and off with emails, voice calls and voice mails.
I use my phone to run my business and obviously this is a huge problem. I've missed very important calls and messages and it's ended up costing me money as well as creating ill feeling between myself and clients. I love my iPhone but to be honest I'm starting to wonder if I made the right choice all those years ago when I switched from BlackBerry. My BlackBerry was absolutely 100% rock solid at these core communication features. Never missed a call or notification. My iPhone is so unreliable with notifications that I'm constantly taking it out of my bag to see if someone's called or texted me.
I've seen this problem reported all over the web and Apple have done nothing to fix it. They cannot be unaware of it. I don't know whether it's a software problem or a hardware problem, but they really need to get it sorted out.