Yes, you should backup your own web hosting files. Even if your web host has their own backup system in place. You should still backup your own files,database, and emails. No backup plan is full proof.
Your web host’s back plans will fail you
Your web host may say that they have a backup system in place but it does not mean that this backup system will be functional when you need it the most. Some web hosting customers have experienced hardships when their web hosts experienced an outage and they were left with no backups. These web hosting customers thought that they didn’t need to backup their files because their web host would do it. This was not the case because there are multiple stories about web hosts that had backup systems that failed or simply didn’t work when disaster struck. Do not rely upon your web host to provide backups or to restore your account if something goes wrong.
Having your own backups helps improve the odds
When you create your own backup copies, it improves the odds that if disaster were to struck then you know one way or other you would have access to your web hosting files. It is better to have too many backup files available rather than to have none available. When you create your own backup files it helps create redundancy if your web host’s backup system for whatever reason does not work. You will know you have your own backup files to work with.
Web host wipes all your files
What would you do if your web host were to “accidentally” wipe all your files? This isn’t a far fetched question but this has occurred. Do not assume or think that your web host would never do such a thing. Your web host may have a lot of different employees and a new employee may hit the wrong key stroke or click the wrong button and “boom” all your files would be gone. Web hosts do make mistakes and technical support will make mistakes. Do not assume your web host won’t “accidentally” delete everything after you open a support ticket about why you can’t install some script.