Yes, if you have network filtering set up for your school's network, the two will act independently. For example, you can use GoGuardian with a standard web filter. Websites blocked by another filter will still be blocked. If a website shows up in GoGuardian’s Flagged Activity section that means it got through the other filter.
If you use two Chrome extension-redirect filters, however, blocking the same URL through both filters can cause issues with filtering. For more information on this, please contact support using the chat bubble in the bottom right corner of the screen.