Apparently not, so much so that Mozilla is now exploring the possibility of introducing a “super private browsing” mode in future versions of Firefox. How this works is by embedding Tor connectivity in Firefox, in which all your internet traffic will be re-routed and encrypted across Tor servers.
According to Mozilla, regular private mode still sends your traffic through your ISP’s servers, or the servers of the company you’re working at, meaning that it isn’t truly as private as you would think. It will also not protect users from malware like keyloggers or spyware, which they think they can guard against by introducing Tor connectivity to Firefox.
The downside to this is that routing traffic through Tor can result in a hit in performance, such as page loading times. This is why Mozilla is exploring the idea to see if this can be done without too much of a hit on performance. Whether or not it will be successful remains to be seen, but it does sound like it holds quite a bit of potential.