I did a live chat with a Samsung agent. They took control of my phone and checked some settings. Ultimately, the agent's solution was to clear the Clock apps cache. I asked him what I should tell someone in case they ask how to fix it. Here is his reply:
"Santino: I have cleared temp files for the clock app on device; However, if the issue persists I recommend you perform wipe cache partition on phone to fix the issue.
Santino: Yes, you can advise them to clear cache and data for the Clock apps under application manager >> All.
Santino: Wipe cache partition will erase all temp files all the applications on the device and will refresh the phones internal circuit.
Santino: I recommend you to perform wipe cache partition if the issue persists.
Santino: I will help you with the steps.
Santino: 1. Power the device off (remove and re-insert the battery).
Santino: 2. Press and hold the Volume Up key and the Home key, then press and hold the Power key.
Santino: 3. When the device vibrates; release only the Power key.
Santino: 4. When the Android system recovery menu displays; release the Volume Up and the Home key.
Santino: 5. Navigate with the Volume keys to the wipe cache partition reset option and press the Power key to select.
Santino: 6. After wipe cache partition, press the Power key to reboot system now.
Santino: This will not erase or effect any data on phone."
When I told this agent that I believe this is a problem with the Galaxy S5, he said they have had no reports of this problem. He also said it's most likely due to 3rd party apps.
Anyway, hopefully this helps other people with this problem. Frankly, I don't even know for sure if this solves my problem! He had me test my alarm and it worked. That tells me nothing. It usually worked, but only about 80% of the time! We'll see if this actually fixes the problem!