I suspect you have one of the following conditions..
1, a conflicting .htaccess file is above the links directory (since you use Joomla, and it includes a .htaccess file with mod-rewrite rules for freindly URLs, this seems likely)
2, The .htaccess file in the links directory is missing.
3, The .htaccess file in the links directory contains the wrong syntax for your version of Apache.
Here is a portion of the FAQ from the old Dew-Code.com site regarding the included .htaccess file
There is a .htaccess file needed in the same directory as the directory file. The .htaccess file is included, but some FTP programs/operating systems have issues with such filenames. The included text file, dothtaccess.txt covers the topic as well.
The contents of the .htaccess file should look like this
<Files directory>
ForceType application/x-httpd-php
#SetHandler application/x-httpd-php
#ForceType x-httpd-php
</Files>
Depending on your version of Apache you may need to use a different syntax by toggling the commented lines in the .htacess file. (note you likely wont see the 2 newer variations in your .htacess file)
If you download/upload the file in order to edit it, be sure it is uploaded in ASCII mode.
If you have access to it, the directives within the .htaccess file can be put in your apache config file, normally /etc/httpd.conf then the .htacess file is not needed.
An httpd.conf entry might look like this...
<VirtualHost>
ServerName
www.mysite.com
DocumentRoot /var/www/html/username
<Directory /var/www/html/username>
Allow from all
AllowOverride All
Order allow,deny
</Directory>
<Files directory>
ForceType application/x-httpd-php
</Files>
</VirtualHost>