Update Issues

I have been trying to install both the new WordPress update to 3.6.1 and updates to my site plugins for a few weeks and am having considerable issues. Every time I input my FTP username and password, the update process begins, but it gets stuck unpacking the update and will never finish the process. With plugins, the site goes into maintenance mode and will not complete either, leaving the site inaccessible for a while.

What could be the issue here?