A few things to check. If the Constant is a site level constant and not a site collection level constant, you must be on the same web level to use it.
You can also set permissions on a constant. Make sure you have valid permissions by going to the constant from the managing link in Site Settings, then click on the constant and choose Edit Permissions in the ribbon. You can change the permissions to be Everyone, Site Collection Admins, or Specific Users.
It also depends whether your constant is sensitive or not. A secure string or credential you can for example not just input in a location where it would show a user the contents of your constant. Next to setting your permissions like Andrew said, check where you are actually trying to use it.