Updated: October 20, 2014
From the little information available in the comment below, I would think Drupal will be just fine for the website. I would simply steer clear of using something like WordPress in such a case. In short, whenever registered users will see different content from visitors, WordPress is just not the right tool (it's a publishing platform, not something to manage user dashboards or user specific content for non-editors of such content).
In addition, Drupal's out of the box user management functionality, among other things, makes it very good for such websites. Drupal also has extensive caching if you need it (and if you don't you can just skip that bit altogether).
You can also use any of the other plethora of CMS out there, but Drupal can certainly be used to accomplish (and sustain and grow) what you seem to be after.
To be clear, Matteo, you really cannot use the same MySQL database unless your database structure can somehow be mapped to Drupal's (and there's a very good chance that is going to be a tedious and painful task).
Different CMS' serve different purposes. What does your website do - how much data do you move between pages? Can you use some kind of caching? How heavy are your pages? Are users registered users or just site visitors?
Drupal is extremely powerful, but it has a learning curve if you're going to get use out of it.