This sounds like an issue with CSS specific to your site or instance - have you tried using FireBug or Chrome Dev Tools (F12 in Chrome) to check which styles are being applied to the accordion? Also, is the number of panes set correctly in the Number of panes property for the Web part? Are you able to reproduce the issue in the sample Corporate site under Examples > Web parts > Layouts > Accordion Layout? I have just tested using 50 panes and they were all displayed correctly and were clickable.
Please let me know if I can be of further assistance.