In addition to upgrading Firefox, upgrade your nss package as well. That's it. On Fedora, you should do:

sudo dnf update nss

On Debian/Ubuntu, you should do:

sudo apt-get update && apt-get install libnss3

Long Story

On a peaceful Sunday afternoon, I decided to upgrade the distro-vendored Firefox browser on Fedora 29. All I had to do was to run this:

sudo dnf update firefox

This should have worked, right? No. Most of the sites I tried browsing gave me a big grey error titled: NS_ERROR_NET_INADEQUATE_SECURITY.

Woah, what is this?

When I was a teenager whom relatives would call at the slightest hint of computer panic, many of the times all I had to do was to either fix their system time or have them replace the motherboard's CMOS battery. My first instinct, thus, was to see if time was alright. And, it was. Also, this is the kind of error you would normally get with incorrect times. Browsers were very helpful with error messages in the past, as they are still now.

A perk of being adult is that your search queries are more succinct. Nowadays, you don't describe your scenario to the browser like "browser not working after upgrade...". You just search the error. That's all. A quick search led me to a similar bug documented at RedHat Bugzilla. Although it was for the older version of Firefox, I tried upgrading nss as suggested in the bug, and voila, it worked. All I had to do was:

sudo dnf update nss

In an ideal world, the updated Firefox package should have referenced to an updated nss package as well. Hopefully, it will be fixed in some future versions. If you're interested in packaging for Linux distributions, it might be useful to dig deeper into this issue and come up with a solution.