When designing your web site, you should consider the portability of your files. The concern might arise if you decide to move your site to a different hosting company. Perhaps you find a better price or you are having service concerns with your current provider.
Any hosting company will support basic files such as HTML, JPEG, and GIF. However they do not all support the same Server Side Scripts. So if your web site is running on ASP, then if you need to move your web site, you will need to find a host that supports that technology. If your site contains several Perl files ( they end with .cgi or .pl) then you will need to find a host that allows you to have those files. Often there are restrictions on where you can put these files, so you may need to redesign your site completely. Also you will have to change your she bang line if you are moving from a Windows Server to Apache or vice versa. If your site has PHP files, you will need to confirm that PHP is installed at the server. If data comes from a MySQL database, then that information will have to be redeveloped.
While you probably won't need to move your site very often, you should be prepared. Since hosting seems like such a low margin business, I would not be surprised to see a reduction in hosting companies in the next few years.