I found that if I power my phone off and back on (press and hold the sleep/wake button on the top until the "Slide to power off" message appears, slide it to power off, wait a minute for it to power off, press and hold the sleep/wake button until the Apple logo appears), alert tones start working again, for a while. Here are the details:
I also have an iPhone 5 with iOS 9.1. I've had this happen twice every 5 or 6 days: I get a calendar alert for an upcoming meeting appear on the screen and the phone vibrates, but the alert tone does not sound. The phone rings normally. The mute switch is not on silent. Volume is up. When I go into settings, sounds, I see that my alert tone (which I bought as a special ringtone via the app store) is checked. When I select another ringtone, it gives me the sample for that ringtone normally, but when I select my custom tone, it just vibrates. I changed the alert tone to another standard tone (Apex) in the list which was working, and that worked for about a day before that tone became silent the same way. Then I had 2 alert tones that were silent. I turned the phone off and back on and now the alert tones worked correctly, and I could select my original custom alert tone and it worked. After about 5 days, the alert tone stopped working again and a power off/on was required to get it working. This is a real problem because I depend on those alerts to get me to my meetings on time. This just started happening in November 2015.