I encountered this same problem, and I looked into the PHP to determine the cause.
You’re logged in as some user who has a lower privilege level than the owner of the blogroll links. Most probably, your blogroll links are owned by admin (privilege level 10) and you’re logged in as a regular user (privilege level <= 9). In this situation, the PHP code refuses to give you the ability to remove blogroll entries.
Solution:
Log out and then log in as ‘admin’. Now if you go to the Links>>ManageLinks page you can either delete the unwanted blogroll items or select them all and “Assign ownership” of them to your less powerful user, log back in as yourself, and delete them (the links will be there).
Hope this helps!