Some CMS, for example Typo3 are rather performance-demanding, even when you do not have much visitors.
There might be, it depends on what you pay for. I guess that somewhere on the website of the hoster there is a description what is included in your hosting package, regarding bandwith, CPU time, RAM.
However: You do not necesseraly need a dedicated server. Perhaps you just need to upgrade your shared hosting package or go to another webhosting provider.
My webhoster for example has different shared hosting packages for different needs. Going from 5€/month to 40€/month. When you choose the 5€ package you will be on a shared server with hundreds of others. You will have a maxiumum of 25 MB RAM for sripts, 10 CPU-seconds, and a maxium script execution time of 30 sec.
With the 40€ package you are on a server with only 8 other websites and have 50 MB RAM for scripts, 20 CPU-sec., max script execution time 90 sec.
If you want to run a CMS like Typo3 on a 5€ webspace - this just won't work. It will however on the 40€ package.
So the answer is: No. You do not necessarily need a dedicated server. You need a webhosting package that suits your needs.