This is not a bug. The applet should run just fine, unless it attempts to do a restricted action. At that point, it will present a security exception with denied permission. What happens after the security exception depends on how the applet is written, but many will continue doing the things they are allowed to do.
Restricted actions include: reading or writing to the local hard drive, printing, contacting a remote server other than the applet host, and accessing shared libraries.