Your Snapchat could be saying that for a number of reasons, as much as developers try to faze out bugs and errors through updates they cannot completely assure that the app will function perfectly for everyone.
The following is a list of reasons why you might be having problems with video recording on Snapchat.
Why Snapchat might say video recording failed
1. The most common fix for this error is freeing up some memory, you're phone needs memory to record and even if its not full this could slow your phone down and cause issues. Make sure you have a good amount of free space enough to record a short video.
2. Make sure that your app is updated to the very latest version to ensure that any bugs may have been fixed.
3. Remove the app from backgrounding, or in essence restart the app. IPhone's and most androids have this backgrounding feature so make sure you try this.
4. If you're app is still giving you an error try restarting your device, the classic "have you tried turning it on and off again?"
5. Try signing out of your Snapchat account and signing back in.
6. If the problem persists further try deleting the app and re-installing it, if that doesn't work try installing an older version, perhaps your device just doesn't like the new version?
If you still can't fix the error then try watching this video. It covers a few things that I have missed: